Supports for minimal hermitian matrices

TL;DR

This paper investigates the structure of supports related to minimal Hermitian matrices, introducing an invariant to measure support proximity and analyzing the geometric properties of the support set within flag manifolds.

Contribution

It defines a new invariant for supports of minimal Hermitian matrices and studies their geometric structure using critical point analysis.

Findings

The invariant δ effectively measures how close subspace pairs are to forming supports.

Supports have interior points in the space of flag manifolds, indicating their richness.

Analysis of the map F provides insights into the structure of supports.

Abstract

We study certain pairs of subspaces $V$ and $W$ of $\mathbb{C}^n$ we call supports that consist of eigenspaces of the eigenvalues $\pm\|M\|$ of a minimal hermitian matrix $M$ ($\|M\|\leq \|M+D\|$ for all real diagonals $D$). For any pair of orthogonal subspaces we define a non negative invariant $\delta$ called the adequacy to measure how close they are to form a support and to detect one. This function $\delta$ is the minimum of another map $F$ defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of $F$ in order to approximate $\delta$. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

Equations

$$\Vert Z\Vert \le \Vert Z+D\Vert ,\text{ for every real diagonal }D,$$

$$Z=\Vert Z\Vert P_{V_{+}}-\Vert Z\Vert P_{V_{-}}+R$$

$$\text{co}\big{(}\{v_{i}\circ\overline{v_{i}}\}_{i=1}^{r}\big{)}\cap\text{co}\left(\{w_{j}\circ\overline{w_{j}}\}_{j=r+1}^{r+s}\right)\neq\emptyset$$

$$\left\{\begin{array}[]{ccc}|v_{1}^{1}|^{2}+|v_{1}^{2}|^{2}+\dotsb+|v_{1}^{p}|^{2}&=&|w_{1}^{1}|^{2}+|w_{1}^{2}|^{2}+\dotsb+|w_{1}^{q}|^{2}\\ |v_{2}^{1}|^{2}+|v_{2}^{2}|^{2}+\dotsb+|v_{2}^{p}|^{2}&=&|w_{2}^{1}|^{2}+|w_{2}^{2}|^{2}+\dotsb+|w_{2}^{q}|^{2}\\ \vdots&\vdots&\vdots\\ |v_{n}^{1}|^{2}+|v_{n}^{2}|^{2}+\dotsb+|v_{n}^{p}|^{2}&=&|w_{n}^{1}|^{2}+|w_{n}^{2}|^{2}+\dotsb+|w_{n}^{q}|^{2}\end{array}\right.$$

$$\begin{array}{rl}{\mathcal{S}}_{(r,s)}=& \{(V,W)\in {\mathbb{C}}^n\times {\mathbb{C}}^n:(V,W)\text{ is a support}\\ & \text{ with }\dim (V)=r\text{ and }\dim (W)=s\}\end{array}$$

$$\begin{array}{rl}{\mathcal{F}}_{(r,s)}=\{(V,W):V\perp W& \text{ are subspaces of }{\mathbb{C}}^n,\\ & \dim (V)=r,\dim (W)=s\}.\end{array}$$

$$\underline{v}=(v^{1},v^{2},\dots,v^{p})=\left(\begin{array}[]{cccc}v_{1}^{1}&v_{1}^{2}&\dots&v_{1}^{p}\\ \vdots&\vdots&\dots&\\ v_{n}^{1}&v_{n}^{2}&\dots&v_{n}^{p}\end{array}\right)$$

$$\sum _{i=1}^p{v}^i\circ \overline{v^i}=\left(|v_1^1{|}^2+\cdots +|v_1^p{|}^2,\dots ,|v_n^1{|}^2+\cdots +|v_n^p{|}^2\right)\simeq \Phi (\underline{v}{\underline{v}}^{\ast })$$

$$⟨\underline{v}{\underline{v}}^{\ast }{e}_i,e_i⟩=⟨x{s}^t{u}^{\ast }{e}_i,x{s}^t{u}^{\ast }{e}_i⟩=\sum _{j=1}^p{s}_j^2|u_i^j{|}^2=\sum _{j=1}^p⟨s_j{u}_i^j,s_j{u}_i^j⟩.$$

$$\tilde{\underline{v}}=(s_1{u}^1,s_2{u}^2,\dots ,s_p{u}^p)$$

$$\Phi (\underline{v}{\underline{v}}^{\ast })=\Phi (\tilde{\underline{v}}{\tilde{\underline{v}}}^{\ast }).$$

$$\widetilde{\underline{v^{\prime }}}=(s_{k_1}{u}^{k_1},s_{k_2}{u}^{k_2},\dots ,s_{k_t}{u}^{k_t})$$

$$\Phi (\underline{v}{\underline{v}}^{\ast })=\Phi (\underline{w}{\underline{w}}^{\ast })$$

$$\sum _{i=1}^p\frac{\Vert v^i{\Vert }^2}{{\sum }_{k=1}^p\Vert v^k{\Vert }^2}\left(\frac{v^i}{\Vert v^i\Vert }\circ \frac{\overline{v^i}}{\Vert v^i\Vert }\right)=\sum _{j=1}^q\frac{\Vert w^j{\Vert }^2}{{\sum }_{k=1}^q\Vert w^k{\Vert }^2}\left(\frac{w^j}{\Vert w^j\Vert }\circ \frac{\overline{w^j}}{\Vert w^j\Vert }\right)$$

$$\Phi (\underline{v}{\underline{v}}^{\ast })=\Phi (\underline{w}{\underline{w}}^{\ast })$$

$$\Phi ({\sigma }_V)\cap \Phi ({\sigma }_W)\ne \varnothing .$$

$$\Phi ({\underline{v}}_k{\underline{v}}_k^{\ast })=\Phi ({\underline{w}}_k{\underline{w}}_k^{\ast })$$

$$\mu (\underline{v})=\frac{i}{2}\underline{v}{\underline{v}}^{\ast }\text{, for}\underline{v}\in M.$$

$$(\mu (\underline{v}){)}_{k,l}=\frac{i}{2}⟨{\underline{v}}_k,{\underline{v}}_l⟩\text{, for }k=1\dots ,n\text{ and }l=1,\dots ,n.$$

$$(\mu (\underline{v}){)}_{k,k}=\frac{i}{2}\Vert {\underline{v}}_k{\Vert }^2\text{, for }k=1,\dots ,n.$$

$${\mu }_d(\underline{v})=\frac{i}{2}\left(\begin{array}{c}\Vert {\underline{v}}_1{\Vert }^2\\ ⋮\\ \Vert {\underline{v}}_n{\Vert }^2\end{array}\right),$$

$$\Phi \left((\alpha \underline{v})(\alpha \underline{v}{)}^{\ast }\right)=|\alpha {|}^2\Phi (\underline{v}{\underline{v}}^{\ast }).$$

$$\Phi (\underline{v}{\underline{v}}^{\ast })=\Phi (\underline{w}{\underline{w}}^{\ast })$$

$$\mathrm{Tr}(\underline{v}{\underline{v}}^{\ast })=1\text{and}\mathrm{Tr}(\underline{w}{\underline{w}}^{\ast })=1.$$

$$S_V\text{ and }{S}_W\text{ the unit spheres of }\mathrm{hom}({\mathbb{C}}^r,V)\text{ and }\mathrm{hom}({\mathbb{C}}^s,W)$$

$$\delta (V,W)=\inf \left\{||\Phi (\underline{v}{\underline{v}}^{\ast })-\Phi (\underline{w}{\underline{w}}^{\ast })|{|}^2:(\underline{v},\underline{w})\in S_V\times S_W\right\}$$

$$\mathcal{V}:{\mathbb{C}}^r\to V,\mathcal{W}:{\mathbb{C}}^s\to W.$$

$${\Sigma }_r=\{a\in M_r^h(\mathbb{C}):\mathrm{Tr}(a^2)=1\}\text{ and }{\Sigma }_s=\{b\in M_s^h(\mathbb{C}):\mathrm{Tr}(b^2)=1\}$$

$$F(a,b)=||\Phi (\mathcal{V}{a}^2{\mathcal{V}}^{\ast })-\Phi (\mathcal{W}{b}^2{\mathcal{W}}^{\ast })|{|}^2$$

$$\delta (V,W)=\underset{(a,b)\in \Sigma }{\min }F(a,b).$$

Full text

For any pair of orthogonal subspaces we define a non negative invariant $\delta$ called the adequacy to measure how close they are to form a support and to detect one. This function $\delta$ is the minimum of another map $F$ defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of $F$ in order to approximate $\delta$. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

1. Introduction

Let $\mathbb{C}^{n}$ denote as usual the vector space of $n$-tuples of complex numbers and $M_{n}(\mathbb{C})$ the $n\times n$ complex matrices. Let $M^{h}_{n}(\mathbb{C})$ (respectively $M^{ah}_{n}(\mathbb{C})$) be the set of hermitian or self-adjoint (respectively anti-hermitian) matrices and $\|\ \|$ the spectral norm in $M_{n}(\mathbb{C})$.

We call $Z\in M^{h}_{n}(\mathbb{C})$ a minimal matrix if

[TABLE]

(a similar definition can be given for antihermitian matrices and pure imaginary diagonals). Minimal matrices allow the description of short length curves in the homogeneous space $\mathcal{P}=U(n)/U(\mathcal{D}_{n})$, where $\mathcal{D}_{n}$ denotes the diagonal $n\times n$ complex matrices, $U(\mathcal{D}_{n})$ the unitary diagonal matrices and $U(n)$ the unitary group of $M_{n}(\mathbb{C})$. More precisely, we consider the homogeneous space $\mathcal{P}$, with the left action $L_{U}(\rho)=U\rho U^{*}$, for $U\in U(n)$, $\rho\in\mathcal{P}$ (where the action is performed on any element of the class $\rho$). Then the space $\mathcal{P}$ is provided with the invariant Finsler metric defined by the quotient norm in $M_{n}(\mathbb{C})^{ah}/\mathcal{D}_{n}^{ah}$, the tangent space of $\mathcal{P}$ at $\rho$. This structure allows the definition of a natural distance $\text{d}(\rho_{1},\rho_{2})$ in $\mathcal{P}$ as the infimum of the length of curves in $\mathcal{P}$ joining $\rho_{1}$ and $\rho_{2}$ (see [2, 3] for details).

The following result is a restatement of Theorem I of [3] in the present context.

Theorem 1**.**

Let $\rho\in\mathcal{P}=U(n)/U(\mathcal{D}_{n})$, $X\in(T\mathcal{P})_{\rho}\simeq M_{n}(\mathbb{C})^{ah}/\mathcal{D}_{n}^{ah}$ and $Z\in M_{n}^{ah}(\mathbb{C})$ a minimal matrix which projects to $X$ ($X=Z\rho-\rho Z$). Then the curve given by $\gamma(t)=L_{e^{tZ}}\rho={e^{tZ}}\rho{e^{-tZ}}$ satisfies $\gamma(0)=\rho$, $\dot{\gamma}(0)=X$ and has minimal length among all curves in $\mathcal{P}$ joining $\gamma(0)$ to $\gamma(t)$ for each $t$ with $|t|\leq\frac{\pi}{2||Z||}$.

Observe that from all the cases covered by Theorem I of [3], the homogeneous space $\mathcal{P}=U(n)/U(\mathcal{D}_{n})$ we are considering here is probably the simplest non commutative non trivial case.

This result motivates the study of minimal matrices in the spectral norm. Some particular properties have been studied already but in the present work we focus on the particular and rich structure of a spectral pair of eigenspaces related to a minimal matrix.

If $Z\in M_{n}^{h}(\mathbb{C})$ is a minimal matrix, then $\pm\|Z\|$ must be eigenvalues of $Z$. Nevertheless, this condition is not enough. If $\pm\|Z\|\in\sigma(Z)$, then $Z$ is minimal if and only if there exist orthogonal corresponding eigenspaces $V_{+}$ and $V_{-}$ (ranges of the spectral projections $P_{V_{+}}$ and $P_{V_{-}}$ corresponding to the eigenvalues $\pm\|Z\|$) that satisfy

[TABLE]

(where $R\in M_{n}^{h}(\mathbb{C})$, its range is orthogonal to $V_{+}\oplus V_{-}$, and $\|R\|\leq\|Z\|$) and such that $V_{+}$ and $V_{-}$ satisfy the following property:

Condition 1. There exist orthonormal sets $\{v_{i}\}_{i=1}^{p}\subset V_{+}$ and $\{w_{j}\}_{j=q+1}^{p+q}\subset V_{-}$ such that

[TABLE]

where $\circ$ denotes the Hadamard or entrywise product and $\text{co}(A)$ the convex hull of $A$ (see Corollary 3 in [1]).

In Theorem 3 it is proved that Condition 1 is equivalent to the following property held by two orthogonal subspaces $V$ and $W$.

Definition 1**.**

Given two orthogonal subspaces $V$ and $W\subset\mathbb{C}^{n}$ we call the pair $(V,W)$ a support if there exist non trivial subsets $\{v^{1},v^{2},\dotsc,v^{p}\}$ of $V$ and $\{w^{1},w^{2},\dotsc,w^{q}\}$ of $W$ with coordinates in the canonical basis given by $v^{i}=(v^{i}_{1},v^{i}_{2},\dotsc,v^{i}_{n})$, for $i=1,\dotsc,p$ and $w^{j}=(w^{j}_{1},w^{j}_{2},\dotsc,w^{j}_{n})$, for $j=1,\dotsc,q$ such that

[TABLE]

or equivalently $\sum_{i=1}^{p}v^{i}\circ\ \overline{v^{i}}=\sum_{j=1}^{q}w^{j}\circ\ \overline{w^{j}}$, where $\circ$ denotes the Hadamard product and $\overline{x}$ the vector formed by the conjugated coordinates of $x$.

This definition can also be stated choosing orthogonal vectors $\{v^{i}\}_{i=1}^{t}$ and $\{w^{j}\}_{j=1}^{h}$ (see Theorem 2).

Remark 1**.**

The previous discussion implies that $Z$ is a minimal matrix with a decomposition as in (1.2) if and only if the pair of subspaces $(V_{+},V_{-})$ is a support (see also Theorem 3).

We will denote with $\mathcal{S}_{(r,s)}$ the set of supports of $\mathbb{C}^{n}$ with corresponding dimensions $r$ and $s$:

[TABLE]

Remark 2**.**

The definition of the set of supports suggests that it might have the structure of a real algebraic set. As expected, $\mathcal{S}_{(r,s)}$ turns out to be closed (see Proposition 2). Nevertheless, the fact that for every $n\in\mathbb{N}_{n\geq 3}$ there exist interior points in $\mathcal{S}_{(r,s)}$ in the ambient of a flag manifold of $\mathbb{C}^{n}$ is a surprising result (see Section 9). It would be interesting to find out if $\mathcal{S}_{(r,s)}$ is a semi-algebraic set.

The previous comments allow us to state the following result.

Remark 3**.**

There exists a function between the set of minimal matrices $Z$ with eigenspaces $V_{+}$ and $V_{-}$ corresponding to the eigenvalues $\pm\|Z\|$ with $\dim(V_{+})=r$, $\dim(V_{-})=s$ onto $\mathcal{S}_{(r,s)}$ that maps $Z$ to the support $(V_{+},V_{-})$.

Consider the equivalence class of a matrix $M\in M_{n}^{h}(\mathbb{C})$ defined by $[M]=\{N\in M_{n}^{h}(\mathbb{C}):(M-N)\in\mathcal{D}_{n}\}$. The relation between $[M]$ and the support determined by its corresponding minimal matrix (or matrices) is a work in progress that will studied elsewhere.

Supports are a fundamental aspect of the description of minimal matrices. In this work we are going to analyze the structure of the set of supports $\mathcal{S}_{(r,s)}$ as a subset of the flag manifold $\mathcal{F}_{(r,s)}$ (see (2.1)) under the identification of $(V,W)\in\mathcal{S}_{(r,s)}$ with $V\oplus W\oplus(\mathbb{C}^{n}\ominus V\ominus W)$. The authors consider that the study of $\mathcal{S}_{(r,s)}\subset\mathcal{F}_{(r,s)}$ is interesting by itself.

In order to measure how far are two subspaces $V$ and $W$ to become a support we define in (5.3) a number $\delta(V,W)\geq 0$ we call the adequacy of $V$ and $W$ that satisfies $\delta(V,W)=0$ if and only if $(V,W)$ is a support. The adequacy is a natural tool to achieve this and can be computed as the minimum of a function $F$ defined on the product of certain spheres $S_{V}\times S_{W}$ of linear maps (see (5.2) and (5.5)). We study the gradient, Hessian and critical points of this $F$ (see Section 5) to allow the approximation of the adequacy. Some of the formulas obtained are used in the appendices to obtain numerical examples of particular supports that are interior points of flag manifolds in low dimensions. These results are used to prove in Theorem 10 that there exist open neighborhoods of supports (formed by supports in $\mathbb{C}^{n}$ for every $n\in\mathbb{N}_{n\geq 3}$) in the flag manifold $\mathcal{F}_{(\dim(V),\dim(W))}$.

We also consider a geometric interpretation of the adequacy in Section 6 describing a new space of parameters to calculate it. This perspective allows the characterization of some critical points of the map $F$ whose global minimum is the adequacy in sections 7 and 8.

2. Preliminaries and notation

Here we introduce some notation used throughout the article. $M_{n}(\mathbb{C})$ will denote the $n\times n$ matrices with coefficients in $\mathbb{C}$, $M_{n}^{h}(\mathbb{C})$ the hermitian matrices and $M_{n}^{ah}(\mathbb{C})$ the anti-hermitian matrices. The expression diag$(a_{1},a_{2},\dots,a_{n})$ denotes the diagonal matrix in $M_{n}(\mathbb{C})$ with the elements $a_{1},a_{2},\dots,a_{n}\in\mathbb{C}$ in its principal diagonal, and $\Phi:M_{n}(\mathbb{C})\to\mathcal{D}_{n}\subset M_{n}(\mathbb{C})$ the conditional expectation such that $\Phi(x)$ is the diagonal matrix formed with the diagonal entries of $x$.

As usual $GL(n,\mathbb{C})$ denotes the general group of invertible matrices in $C^{n\times n}$. And $\operatorname{Gr}(k,\mathbb{C}^{n})$ will denote the Grassmannian manifold of all $k$-dimensional subspaces of $\mathbb{C}^{n}$.

We denote with $v\circ w$ the Hadamard (or Schur) entrywise product of two vectors $v$, $w\in\mathbb{C}^{n}$, where $v\circ w\in\mathbb{C}^{n}$, and $(v\circ w)_{i}=v_{i}w_{i}$, for $i=1,\dots n$. Similarly $A\circ B$ is the Hadamard (or Schur) product of two matrices $A,B\in M_{n}(\mathbb{C})$.

We use $\mathcal{F}_{(r,s)}$ to represent the set

[TABLE]

Observe that the pair $(V,W)\in\mathcal{F}_{(r,s)}$ can be identified with the element $\{0\}\subset V\subset V\oplus W\subset\mathbb{C}^{n}$ in a classic flag manifold $F(r,r+s,n)$ which is isomorphic to the homogeneous space U(n)/\big{(}U(r)\times U(s)\times U(n-r-s)\big{)}. Therefore, $\mathcal{F}_{(r,s)}$ can be identified with the flag manifold $F(r,r+s,n)$.

3. Properties of a support $(V,W)$

Given $v^{1},v^{2},\dotsc,v^{p}\in V$, for $V$ a subspace of $\mathbb{C}^{n}$, we denote with

[TABLE]

either the $n\times p$ matrix or the $\mathbb{C}^{p}\to\mathbb{C}^{n}$ linear map. Let $\Phi:M_{n}(\mathbb{C})\to\mathcal{D}_{n}$ be the map such that $\Phi(x)$ is the diagonal of $x$. Then the $n$-tuples that appear in (1.4) can be written

[TABLE]

where we identified the vector with the diagonal matrix $\Phi(\underline{v}\ \underline{v}^{*})$ of $\underline{v}\ \underline{v}^{*}\in M^{h}_{n}(\mathbb{C})$.

Then using the singular value decomposition of $\underline{v}=usx^{*}$, with $u\in U(n)$, $x\in U(p)$ and $s$ the $n\times p$ diagonal matrix of the singular values of $\underline{v}$ in the $s_{j,j}$ entries. Let us denote them with $s_{j}$, $j=1,\dots,p$. Now consider the column vectors $u^{i}\in\mathbb{C}^{n}$, $i=1,\dots,n$, from the unitary matrix $u$. Note that these $u^{i}$ are eigenvectors of $\underline{v}\ \underline{v}^{*}$.

Let $e_{i}$, for $i=1,\dots,n$, be the $i^{\text{th}}$ element of the canonical basis of $\mathbb{C}^{n}$. Then the $i^{\text{th}}$ diagonal elements of $\Phi(\underline{v}\ \underline{v}^{*})$ are

[TABLE]

Therefore if we consider the $n\times p$ matrix given by its columns

[TABLE]

then the computation made in (3.3) proves that

[TABLE]

Moreover, these columns generate the same subspace than the original $\{v^{j}\}_{j=1}^{p}$.

Let $K\subset\{1,\dots,p\}$ be the subset of indexes such that $s_{k}\neq 0$ if and only if $k\in K$ and let $t=\#(K)$. Then the vectors $\{s_{k}u^{k}\}_{k\in K}\subset\mathbb{C}^{n}$ are orthogonal to each other and generate the same subspace than the original columns $v^{j}$ of $\underline{v}$ for $j=1,\dots,p$.

Therefore, if we consider the $n\times t$ matrix with columns $s_{k}u^{k}$, $k\in K$

[TABLE]

then its columns form an orthogonal basis of the subspace generated by $\{v^{j}\}_{j=1}^{r}$ and it is apparent that also $\Phi\left(\widetilde{\underline{v^{\prime}}}\ \widetilde{\underline{v^{\prime}}}^{*}\right)=\Phi(\widetilde{\underline{v}}\ \widetilde{\underline{v}}^{*})=\Phi(\underline{v}\ \underline{v}^{*})$.

Therefore, we have proved the following result.

Theorem 2**.**

If $(V,W)$ is a support in $\mathbb{C}^{n}$ as in Definition 1 then there exists not null orthogonal vectors $\{v^{i}\}_{i=1}^{t}\subset V$ and $\{w^{j}\}_{j=1}^{h}\subset W$ that satisfy equation (1.4), or equivalently $\sum_{i=1}^{t}v^{i}\circ\ \overline{v^{i}}=\sum_{j=1}^{h}w^{j}\circ\ \overline{w^{j}}$.

Remark 4**.**

Observe that in Definition 1 the vectors $\{v^{i}\}_{i=1,\dots,r}$ of the subspace $V$ are not required to be linearly independent nor generators of $V$, and similarly for $\{w^{j}\}_{j=1,\dots,s}$ in $W$, but the previous theorem states that orthonormal vectors can be chosen. Moreover, these vectors can be taken bounded in norm with a fixed constant $C$ after multiplying all of them by $\frac{C}{\|x\|}$ where $\|x\|$ is the greatest norm of all the vectors considered.

Definition 2**.**

If $\{v^{1},\dotsc,v^{p}\}$ is a system of $p$ vectors in $\mathbb{C}^{n}$, then the diagonal matrix (or corresponding vector) $\Phi(\underline{v}\ \underline{v}^{*})$ will be called the moment of the system $\{v^{1},\dotsc,v^{p}\}$ (with the notation of $\underline{v}\in\mathbb{C}^{n\times p}$ detailed in (3.1)).

Therefore the previous discussion also proved the following result.

Proposition 1**.**

If $\{v^{1},\dotsc,v^{p}\}$ is a system of $p$ linearly independent vectors in $V$, then there is an orthogonal basis $\{c^{1},\dotsc,c^{p}\}$ of $V$ with the same moment than that of $\{v^{1},\dotsc,v^{p}\}$.

Remark 5**.**

Observe that if for $V\perp W$ subspaces of $\mathbb{C}^{n}$, there exist $\{v^{1},\dots,v^{p}\}\subset V$, $\{w^{1},\dots,w^{q}\}\subset W$, and we define $\underline{v}\in\mathbb{C}^{n\times p}$ and $\underline{w}\in\mathbb{C}^{n\times q}$ as in (3.1), then the equality

[TABLE]

is equivalent to the fact that $(V,W)$ is a support in $\mathbb{C}^{n}$ (see (3.2)).

Given a support $(V,W)$ of $\mathbb{C}^{n}$ Proposition 1 implies that there exists an orthogonal set $\left\{{v^{i}}\right\}_{i=1}^{p}$ for $V$ and $\left\{{w^{j}}\right\}_{j=1}^{q}$ for $W$ that satisfy (1.4). Now consider the orthonormal corresponding set after normalizing each vector. Now adding all the equations in (1.4) we obtain that $\sum_{i=1}^{p}\|v^{i}\|^{2}=\sum_{j=1}^{q}\|w^{j}\|^{2}$, and then

[TABLE]

which in turns implies that Condition 1 stated in (1.3) holds. Then statement (b) of Corollary 3 in [1] is fulfilled and $M=\lambda\ P_{V}-\lambda\ P_{W}+R\in M^{h}_{n}(\mathbb{C})$ (with $P_{V}R=P_{W}R=0$, $R\in M_{n}^{h}(\mathbb{C})$, $\|R\|\leq\lambda>0$) is a minimal matrix in the sense that $\|M\|\leq\|M+D\|$ for all real diagonal matrices $D\in M^{h}_{n}(\mathbb{C})$ and $\|\ \|$ the spectral norm (see [1]). Then a support allows the construction of a minimal matrix, and vice versa. In the following theorem we collect some statements that are equivalent to the definition of a support.

Theorem 3**.**

Let $V,W$ be two non trivial orthogonal subspaces of $\mathbb{C}^{n}$, then the following statements are equivalent.

1. (1)

$(V,W)$* is a support, that is, there exist non trivial subsets $\{v^{1},v^{2},\dotsc,v^{p}\}$ of $V$ and $\{w^{1},w^{2},\dotsc,w^{q}\}$ of $W$ such that (1.4) holds.* 2. (2)

The hermitian matrix $M=\lambda(P_{V}-P_{W})+R$ is minimal (see (1.1)) for every $\lambda\in\mathbb{R}_{>0}$, $R\in M_{n}^{h}(\mathbb{C})$, $\|R\|\leq\lambda$, $R(P_{V}+P_{W})=0$. 3. (3)

There exist non trivial subsets $\{v^{1},v^{2},\dotsc,v^{p}\}$ of $V$ and $\{w^{1},w^{2},\dotsc,w^{q}\}$ of $W$ such that

[TABLE]

with $\underline{v}$ and $\underline{w}$ defined as in (3.1) and $\Phi(m)$ the diagonal of $m$. 4. (4)

The sets $\sigma_{V}=\{c\in M_{n}^{h}(\mathbb{C}):P_{V}c=c\geq 0,\ \operatorname{Tr}(c)=1\}$ and $\sigma_{W}=\{d\in M_{n}^{h}(\mathbb{C}):P_{W}d=d\geq 0,\ \operatorname{Tr}(d)=1\}$, satisfy

[TABLE]

Proof.

1. $\Leftrightarrow$ 2. follows after the previous discussion.

2. $\Leftrightarrow$ 4. is proved using the comments following the proof of Corollary 3 in [1] or the property mentioned in (6.3).

3. $\Leftrightarrow$ 3. is Remark 5. ∎

Proposition 2**.**

The set of supports $\mathcal{S}_{(r,s)}$ is closed in the flag manifold $\mathcal{F}_{(r,s)}$.

Proof.

Consider a sequence of pairs of supports given by $\{(V_{k},W_{k})\}_{k\in\mathbb{N}}\subset\mathcal{S}_{(r,s)}$ and such that its corresponding orthogonal projections converge. It is apparent that there exist $V$ and $W$ subspaces of $\mathbb{C}^{n}$ such that $\dim(V)=r$, $\dim(W)=s$, $V\perp W$ and satisfy $\lim_{k\to\infty}P_{V_{k}}=P_{V}$ and $\lim_{k\to\infty}P_{W_{k}}=P_{W}$, that is, $(V,W)\in\mathcal{F}_{(r,s)}$. We only need to prove that the condition (1.4) holds.

Consider for each pair $(V_{k},W_{k})$ a pair of matrices $(\underline{v}_{k},\underline{w}_{k})$ that satisfy

[TABLE]

as in Remark 5.

Note that as mentioned in Remark 4 we can choose the column vectors of the matrices $\underline{v}_{k}$ and $\underline{w}_{k}$ with norm less or equal than one. Then using compacity arguments and after taking subsequences we can suppose that the matrices $\underline{v}_{k}$ are of the same size, and their columns converge to vectors in $V$ that form a matrix $\underline{v}$. Similar arguments can be used for $\underline{w}_{k}$ to obtain a matrix $\underline{w}$. Since for all $k$ equality (3.8) holds, then $\lim_{k\to\infty}\Phi(\underline{v}_{k}\ \underline{v}_{k}^{*})=\lim_{k\to\infty}\Phi(\underline{w}_{k}\ \underline{w}_{k}^{*})$ which is $\Phi(\underline{v}\ \underline{v}^{*})=\Phi(\underline{w}\ \underline{w}^{*})$. Since this is equivalent to the equalities (1.4) (see Remark 5) then $(V,W)$ is a support. ∎

4. Symplectic interpretation of the map $\Phi$

Consider the manifold $M=(\mathbb{C}^{n})^{r}$ composed of matrices $\underline{v}$ defined in (3.1). We denote by $\underline{v}_{k}$, $k=1,\ldots,n$ the rows of $\underline{v}$ considered as vectors in $\mathbb{C}^{r}$.

Since $\mathbb{C}^{n}$ carries a natural symplectic form, so does $M$ (the product form). In this way, $M$ becomes a symplectic manifold. We consider next the left operation action of the unitary group $U(n)$ on $M$. This operation is symplectic. Now we identify the Lie algebra $u(n)$ of $U(n)$ with its dual $u^{*}(n)$ using the inner product $\langle A,B\rangle=tr(AB^{*}).$

In this context the moment map $\mu:M\to u^{*}(n)$ can be computed explicitly:

[TABLE]

Observe that the entries of the matrix $\mu(\underline{v})$ are

[TABLE]

and the entries of the diagonal are

[TABLE]

Finally, observe that for the induced left action of the diagonal unitary matrices $\mathbb{T}^{n}\subset U(n)$ on $M$, the corresponding moment map $\mu_{d}$ is obtained as follows $\mu_{d}:M\to(t^{n})^{*}$

[TABLE]

which is exactly $i/2$ times what was called the moment of the system $\{v^{1},\ldots,v^{r}\}$ in Definition 2.

5. Adequacy of a pair of orthogonal subspaces

Recall that with ${\mathcal{F}}_{(r,s)}$ we denote the space pairs of orthogonal subspaces $(V,W)$ of $\mathbb{C}^{n}$ with $r=\dim V$ and $s=\dim W$. Also consider systems $\underline{v}=(v^{1},\ldots,v^{r}):\mathbb{C}^{r}\to C^{n}$ as in (3.1) and similarly $\underline{w}=(w^{1},\ldots,w^{s}):\mathbb{C}^{s}\to C^{n}$ of vectors in $\mathbb{C}^{n}$ such that $\operatorname{Im}(\underline{v})\subset V$ and $\operatorname{Im}(\underline{w})\subset W$.

Recall that in Definition 2 we called $\Phi(\underline{v}\ \underline{v}^{*})$ the moment of the system $\underline{v}$ where $\Phi$ is the conditional expectation that associates to any $n\times n$ matrix its diagonal part. This map takes its values in the subalgebra $\mathcal{D}_{n}$ of diagonal $n\times n$ matrices which will sometimes be identified with $\mathbb{C}^{n}$. Observe that the map $\underline{v}\mapsto\Phi(\underline{v}\ \underline{v}^{*})$ is homogeneous in the following sense:

[TABLE]

Recall that with this notation $(V,W)$ is called a support (see 1.4 and Theorem 2 and Proposition 1) if there is a non trivial pair $(\underline{v},\underline{w})$ with $\operatorname{Im}(\underline{v})\subset V$, $\operatorname{Im}(\underline{w})\subset W$ such that

[TABLE]

(here non trivial refers to $\underline{v}\neq(0,\ldots 0)$ and $\underline{w}\neq(0,\ldots 0)$).

Observe that if there is a non trivial pair $(\underline{v},\underline{w})$ as before such that $\Phi(\underline{v}\ \underline{v}^{*})$ and $\Phi(\underline{w}\ \underline{w}^{*})$ are only linearly dependent then choosing $\alpha\in\mathbb{R}$ appropriately we can get $\Phi\left((\alpha\underline{v})(\alpha\underline{v})^{*}\right)=\Phi(\underline{w}\ \underline{w}^{*})$, with $\operatorname{Im}(\alpha\underline{v})\subset V$ so that $(V,W)$ is a support.

The objective of this section is to define and compute a “numerical obstruction” for the pair $(V,W)$ to be a support, i.e. a non negative invariant of $(V,W)$ which vanishes if and only if the pair $(V,W)$ is a support. We will call this obstruction the adequacy of $(V,W)$.

Note that if (1.4) holds for the vector columns of $\underline{v}$ and $\underline{w}$ then $\operatorname{Tr}(\underline{v}\ \underline{v}^{*})=\operatorname{Tr}(\underline{w}\ \underline{w}^{*})$ follows. Then the remark made in (5.1) about the homogeneous nature of $\omega$ allow us to restrict to the space of pairs $(\underline{v},\underline{w})$ that are “normalized” in the sense that

[TABLE]

Observe that in the space $\hom(\mathbb{C}^{r},V)$ we have a natural norm given by $\operatorname{Tr}(\underline{v}\ \underline{v}^{*})^{1/2}$ and the same holds for $\hom(\mathbb{C}^{s},W)$. Therefore if we denote with

[TABLE]

respectively, then the selected pairs $(\underline{v},\underline{w})$ belong to $S_{V}\times S_{W}$.

Finally we define the adequacy of the pair $(V,W)$.

Definition 3**.**

Given a pair of non trivial orthogonal subspaces $V,W\subset\mathbb{C}^{n}$, its adequacy is defined as the number

[TABLE]

with $S_{V}$ and $S_{W}$ defined in (5.2).

Since $S_{V}\times S_{W}$ is compact there always exist $(\underline{v},\underline{w})$ in $S_{V}\times S_{W}$ such that $\delta(V,W)$ is attained. Note that $\delta(V,W)=0$ implies that the subspaces $V$ and $W$ form a support (see Definition 1).

Next, in order to compute $\delta(V,W)$ we introduce convenient parameters.

• •

First we fix two isometries

[TABLE]

Observe that in particular, $P_{V}=\mathcal{V}\mathcal{V}^{*}$ and $P_{W}=\mathcal{W}\mathcal{W}^{*}$ are the orthogonal projections in $\mathbb{C}^{n}$ onto $V$ and $W$ respectively.

• •

Then any morphism $f:\mathbb{C}^{r}\to V$ is of the form $f=\mathcal{V}g$ for $g:\mathbb{C}^{r}\to\mathbb{C}^{r}$ a linear map. If we write the polar form $g=au$ where $a\geq 0$ and $u$ is unitary, we have $f=\mathcal{V}au$. Therefore we observe, in relation to the problem of parametrization:

1. (1)

$\operatorname{Tr}(ff^{*})=\operatorname{Tr}(\mathcal{V}a^{2}\mathcal{V}^{*})=\operatorname{Tr}(\mathcal{V}^{*}\mathcal{V}a^{2})=\operatorname{Tr}(a^{2})$ so that $f\in S_{V}$ if and only if $\operatorname{Tr}(a^{2})=1$. 2. (2)

And we have $\Phi(ff^{*})=\Phi(\mathcal{V}a^{2}\mathcal{V}^{*})$.

Similar considerations can be done for $\mathcal{W}$ and $S_{W}$.

In view of these remarks we parametrize the problem of finding the minimum of $\delta(V,W)$ as follows.

The parameter space will be $\Sigma=\Sigma_{r}\times\Sigma_{s}\ \text{, where }$

[TABLE]

are the unit spheres of the self-adjoint matrices (positive or not) of sizes $r\times r$ and $s\times s$.

The function we have to minimize is $F:\Sigma\to[0,+\infty)$, defined by

[TABLE]

where the norm is given by $\|x\|=\sqrt{Tr(x^{*}x)}$. Its minimum value is the adequacy

[TABLE]

In the next computations, in order to alleviate the notation, we will write

[TABLE]

5.1. The gradient of $F$

Now we let $a$ vary as a function of a real parameter $t$ and independently $b$ vary as a function of $u$. Then

[TABLE]

If we denote with $\frac{da}{dt}=X\ ,\ \frac{db}{du}=Y$, then

[TABLE]

Here the inner products are traces of products, so using that $\Delta$ is diagonal we can write

[TABLE]

Therefore

[TABLE]

where the inner products now involved are the natural ones in $r\times r$ and $s\times s$ matrices using the corresponding traces.

If in these algebras $M_{r}(\mathbb{C})$ and $M_{s}(\mathbb{C})$ we consider the operators

[TABLE]

then its adjoints (for the natural inner products) are precisely $S_{a^{*}}=S_{a}$ and $S_{b^{*}}=S_{b}$ since $a,b$ are self-adjoint. So we can write

[TABLE]

Therefore we obtained the following result.

Theorem 4**.**

The gradient of the function $F:\Sigma\to[0,+\infty)$, $F(a,b)=||\Phi(\mathcal{V}a^{2}\mathcal{V}^{*})-\Phi(\mathcal{W}b^{2}\mathcal{W}^{*})||^{2}$ on the riemannian manifold $\Sigma=\Sigma_{r}\times\Sigma_{s}$ at $(a,b)$ (with $\Sigma_{r}$ and $\Sigma_{s}$ as in (5.4)) is

[TABLE]

where the subscript “$tan$” refers to the tangential component (to the sphere $\Sigma_{r}\times\Sigma_{s}$) of the corresponding vector: $X_{tan}=X-\langle X,a\rangle a$, for $X\in M_{r}(\mathbb{C})$ and $Y_{tan}=Y-\langle Y,b\rangle b$, for $Y\in M_{s}(\mathbb{C})$, $\Delta$ is defined in (5.7), $S_{a}$, $S_{b}$ in (5.8) and $\mathcal{V}$, $\mathcal{W}$ are fixed isometries as in (• ‣ 5).

5.2. Approximation of the adequacy $\delta(V,W)$.

The previous theorem allow us to construct a gradient descent type algorithm to approximate the adequacy of a pair of orthogonal subspaces $(V,W)$ in $\mathbb{C}^{n}$.

1. (1)

Starting with $V$ and $W$, construct the corresponding isometries $\mathcal{V}\in\mathbb{C}^{n\times r}$ and $\mathcal{W}\in\mathbb{C}^{n\times s}$ defined in (• ‣ 5) (take for example an orthonormal basis of $V$ and build the matrix $\mathcal{V}$ whose columns are the vectors of that basis, similarly for $\mathcal{W}$). 2. (2)

Choose randomly two positive definite trace one matrices $a_{1}\in\mathbb{C}^{r\times r}$ and $b_{1}\in\mathbb{C}^{s\times s}$. 3. (3)

Then for $i=1,\dots,k$ calculate recursively:

1. (a)

$(a_{i},b_{i})-\operatorname{grad}_{(a_{i},b_{i})}F$ using the identity 5.9:

[TABLE]

where $S_{c}(X)=cX+Xc$, and $\Delta_{i}=\Phi(\mathcal{V}a_{i}^{2}\mathcal{V}^{*})-\Phi(\mathcal{W}b_{i}^{2}\mathcal{W}^{*})$. 2. (b)

Then consider $(\alpha_{i+1},\beta_{i+1})=(a_{i},b_{i})-\operatorname{grad}_{(a_{i},b_{i})}F$, and define $a_{i+1}$ and $b_{i+1}$ as its modules with unit norm:

[TABLE] 3. (c)

If $i+1<k$ go back to step a) and continue the iteration with $a_{i+1}$ and $b_{i+1}$. 4. (4)

After finishing the $k$ iterations compute $\operatorname{Tr}(\Delta_{k+1}\Delta_{k+1})$ to approximate the adequacy $\delta(V,W)$ (see 5.5).

In Figure 1 it is shown the output of several evaluations of the adequacy using the previous procedure on a pair of orthogonal subspaces moved with the multiplication of a curve of unitary matrices.

Remark 6**.**

Some of the examples presented in A, B and C were obtained using the previous algorithm to approximate the adequacy.

5.3. The critical points of $F$

The point $(a,b)\in\Sigma$ is critical for $F$ if and only if $S_{a}(\mathcal{V}^{*}\Delta\mathcal{V})$ is normal to $\Sigma_{r}$ and $S_{b}(\mathcal{W}^{*}\Delta\mathcal{W})$ is normal to $\Sigma_{s}$. Then we can state the following result.

Theorem 5**.**

The point $(a,b)\in\Sigma=\Sigma_{r}\times\Sigma_{s}$ is critical for $F$ if and only if

[TABLE]

5.4. Analysis of the conditions (5.10)

Suppose that we have operators $c\geq 0$, $u$ self-adjoint and $cu+uc=\eta c$ where $\eta\in\mathbb{R}$. Then the following commutation rule holds

[TABLE]

Then, $u$ commutes with $c$ and we have $uc=cu=\frac{\eta}{2}c$. The previous comments allow us to state the next result.

Theorem 6**.**

In a critical point $(a,b)$ of $F$ as in (5.10) where $a\geq 0$ and $b\geq 0$ then $\mathcal{V}^{*}\Delta\mathcal{V}$ commutes with $a$ and $(\mathcal{V}^{*}\Delta\mathcal{V})a=\frac{\lambda}{2}a$ and also $\mathcal{W}^{*}\Delta\mathcal{W}$ commutes with $b$ and $(\mathcal{W}^{*}\Delta\mathcal{W})b=\frac{\mu}{2}b.$

Remark: In these notes we are interested in the minimum value of $F$ on $\Sigma$. Since $(a,b)\in\Sigma$ implies $(|a|,|b|)\in\Sigma$, because $a^{2}=|a|^{2}$, $b^{2}=|b|^{2}$ if $a,b$ are hermitian, and $F(a,b)=F(|a|,|b|)$ it is clear that the minimum of $F$ is attained on some $(a,b)$ with $a\geq 0$ and $b\geq 0$.

5.5. The Hessian of the map $F$

Recall the expression of $\operatorname{grad}_{(a,b)}F$ obtained in (5.9) and the definition of $S_{a}$ and $S_{b}$ in (5.8) for $a\in\Sigma_{r}$ and $b\in\Sigma_{s}$ (see (5.4)).

We write $V_{tan}$ to denote the tangential part of $V\in M^{h}_{r}(\mathbb{C})$ of the sphere $\Sigma_{r}$ when $V$ is considered as a tangent vector at a point of $\Sigma_{r}$ (correspondingly for $W\in M^{h}_{s}(\mathbb{C})$ and $\Sigma_{s}$).

Let us denote with $\pi_{r}:M^{h}_{r}(\mathbb{C})\to T(\Sigma_{r})_{a}$ and $\pi_{s}:M^{h}_{s}(\mathbb{C})\to T(\Sigma_{s})_{b}$

[TABLE]

Recall also that in a riemannian manifold, the Hessian of a function $U$ at a critical point is given by

[TABLE]

where $D$ denotes the covariant derivative of the Levi-Civita connection of the metric. Finally recall that the covariant derivative in our case is the tangent projection of the “ambient” derivative.

In the computations below we will need expressions for the derivatives $\partial_{X}$ and $\partial_{Y}$ in the directions $X\in T(\Sigma_{r})_{a}$ and $Y\in T(\Sigma_{s})_{b}$ respectively of the projections $\pi_{r}$ and $\pi_{s}$.

Recall that in (5.9) we calculated

[TABLE]

where $\Delta=\Phi(\mathcal{V}a^{2}\mathcal{V}^{*})-\Phi(\mathcal{W}b^{2}\mathcal{W}^{*}).$ In order to calculate (5.13) we can use that $D_{(X,Y)}=D_{(X,0)+(0,Y)}=D_{(X,0)}+D_{(0,Y)}.$

Then the covariant derivative of $\pi_{r}(S_{a}(\mathcal{V}^{*}\Delta\mathcal{V}))$ is given by

[TABLE]

where we have used that $S_{a}$ and $S_{X}$ are self-adjoint and $S_{a}(X)=S_{X}(a)$.

The covariant derivative of $\pi_{s}(S_{b}(\mathcal{W}^{*}\Delta\mathcal{W}))$ can be calculated similarly.

Observe that

[TABLE]

and $\partial_{Y}(\Delta)=-\Phi(\mathcal{W}S_{b}(Y)\mathcal{W}^{*})$. Then using that $\partial_{X}\Delta+\partial_{Y}\Delta$ is diagonal

[TABLE]

where we have used in the last equality the formula obtained in (5.15) for $\partial_{X}\Delta$. Similarly we can prove that $\langle S_{b}(\mathcal{W}^{*}(\partial_{X}\Delta+\partial_{Y}\Delta)\mathcal{W}),Y\rangle=-\langle(\partial_{X}\Delta+\partial_{Y}\Delta),\partial_{Y}\Delta\rangle$. Then

[TABLE]

Finally, using the expression (5.13) for the quadratic form $H(F)\left((X,Y),(X,Y)\right)$ and (5.14) we obtain that

[TABLE]

where we have used that $a\perp X$, $b\perp Y$, $\langle S_{X}(\mathcal{V}^{*}\Delta\mathcal{V}),X\rangle=2\langle\mathcal{V}^{*}\Delta\mathcal{V}X,X\rangle$ and $\langle S_{Y}(\mathcal{W}^{*}\Delta\mathcal{W}),Y\rangle=2\langle\mathcal{W}^{*}\Delta\mathcal{W}Y,Y\rangle$. We could simplify the expression of the Hessian even more using that $\langle S_{a}(\mathcal{V}^{*}\Delta\mathcal{V}),a\rangle=2\langle\mathcal{V}^{*}\Delta\mathcal{V},a^{2}\rangle$ and $\langle S_{b}(\mathcal{W}^{*}\Delta\mathcal{W}),b\rangle=2\langle\mathcal{W}^{*}\Delta\mathcal{W},b^{2}\rangle$ to obtain the following result

Theorem 7**.**

The Hessian of the map $F:\Sigma_{r}\times\Sigma_{s}\to\mathbb{R}_{\geq 0}$ (see (5.5) and (5.4)) for $X\in T(\Sigma_{r})_{a}$ and $Y\in T(\Sigma_{s})_{b}$ at a critical point $(a,b)$ can be calculated as

[TABLE]

6. A geometric interpretation of the adequacy

Let $V$ and $W$ be two orthogonal subspaces of $\mathbb{C}^{n}$ with dim$(V)=r$ and dim$(W)=s$ as before.

In this section we distinguish three subalgebras of $M_{n}(\mathbb{C})$.

1. (1)

$\mathcal{D}_{n}\subset M_{n}(\mathbb{C})$ the subalgebra of diagonal matrices and $\Phi:M_{n}(\mathbb{C})\to\mathcal{D}_{n}$ the conditional expectation that associates to the matrix $m$ its diagonal part $\Phi(m)$ as before. Observe that $\Phi$ is an orthogonal projection for the natural Hilbert structure of $M_{n}(\mathbb{C})$. We have for $m\in M_{n}(\mathbb{C})$

[TABLE]

where $p_{k}$ is the orthogonal projection onto the $k$-axis of $\mathbb{C}^{n}$. 2. (2)

We denote with $M_{n}(V)\subset M_{n}(\mathbb{C})$ the subalgebra of the endomorphisms $x$ of $\mathbb{C}^{n}$ which commute with $P_{V}=\mathcal{V}\mathcal{V}^{*}$ (for $\mathcal{V}:\mathbb{C}^{r}\to V\subset\mathbb{C}^{n}$ an isometry with range $V$) and verify $P_{V}x=x$. Observe that $M_{n}(V)$ is a C∗-subalgebra of $M_{n}(\mathbb{C})$ with identity $P_{V}$.

Also the map $\mathcal{P}_{V}:M_{n}(\mathbb{C})\to M_{n}(\mathbb{C})$ defined by

[TABLE]

satisfies the requirements of a conditional expectation in $M_{n}(\mathbb{C})$ with image $M_{n}(V)$, except for the fact that $\mathcal{P}_{V}(I)=P_{V}\neq I$. Finally

[TABLE]

defines an isomorphism of C∗-algebras between $M_{r}(\mathbb{C})$ and $M_{n}(V)$. 3. (3)

Similarly we denote $M_{n}(W)$, $P_{W}$, $\mathcal{P}_{W}$ and $\mathcal{I}_{W}$ related to the subspace $W$. Notice that $M_{n}(V)$ and $M_{n}(W)$ are orthogonal in $M_{n}(\mathbb{C})$ for the Hilbert space structure and also in the sense that

[TABLE]

Now we analyze the optimization problem of computing the adequacy of $(V,W)$ in this context.

We denote with $M^{h}_{n}(V)$ the self-adjoint part of $M_{n}(V)$. The function $a\mapsto\mathcal{V}a^{2}\mathcal{V}^{*}$ maps bijectively the positive part $\Sigma_{r}^{+}$ of the unit sphere $\Sigma_{r}=\{a\in M_{r}^{h}(\mathbb{C}):\operatorname{Tr}(a^{2})=1\}$ (see (5.4)) onto the set

[TABLE]

Note that if $c\in\sigma_{V}$, then $\mathcal{V}^{*}c^{1/2}\mathcal{V}$ lies in $\Sigma_{r}^{+}$, where $c^{1/2}$ is the positive square root of the operator $c$. Similar considerations apply to $W$ and we can define the corresponding $\sigma_{W}$.

Recall that the minimum of the function $F$ (the adequacy of the pair $(V,W)$, see (5.6)) is attained, among other points, at some $(a,b)\in\Sigma_{r}\times\Sigma_{s}$ where $a\geq 0$ and $b\geq 0$. Therefore the adequacy can be obtained as the square of the distance of the set $\Phi(\sigma_{V})$ to the set $\Phi(\sigma_{W})$

[TABLE]

Now we describe the set $\Phi(\sigma_{V})$ (and similarly $\Phi(\sigma_{W})$). Clearly $\sigma_{V}$ is a convex compact set in $M^{h}_{n}(V)$ and therefore $\sigma_{V}$ is the convex hull of the set $\sigma_{V}^{e}$ of its extremal points. Since $\mathcal{I}_{V}:M_{r}(\mathbb{C})\to M_{n}(V)\subset M_{n}(V)$ is an isomorphism of C∗-algebras, the set $\sigma_{V}^{e}$ consists of the projections $p$ of rank one in $M_{n}(V)$. Now these projections $p$ are obtained as follows

[TABLE]

In this case the diagonal of $p$ coincides with $\Phi(p)=\text{diag}\left(|u_{1}|^{2},\dots,|u_{n}|^{2}\right).$

Let us denote by $\Sigma_{V}\subset\mathbb{C}^{n}$ the unit sphere of $V$ and correspondingly by $\Sigma_{W}\subset\mathbb{C}^{n}$ the unit sphere of $W$.

Also define $m:\mathbb{C}^{n}\to\mathbb{R}^{n}$ by

[TABLE]

where we identified the diagonal $\Phi(vv^{*})$ with the vector $((vv^{*})_{1,1},\dots,(vv^{*})_{n,n})\in\mathbb{R}^{n}_{\geq 0}$. Then we can state the next result.

Theorem 8**.**

If $m$ is as in (6.4), $\Sigma_{V}$ is the unit sphere of the subspace $V$ and $\mathop{co}(m(\Sigma_{V}))$ is the convex hull of the set $m(\Sigma_{V})$, then

[TABLE]

for $\Phi$ defined in (6.1) and $\sigma_{V}$ in (6.2).

Proof.

Since $\Phi$ is linear, $\Phi(\sigma_{V})$ is a convex compact set in $\mathbb{R}^{n}$. Therefore, $\Phi(\sigma_{V})$ is the convex hull of its extremal set. But it is well known that the extremal set of $\Phi(\sigma_{V})$ is contained in the image $\Phi(\sigma_{V}^{e})$ which is $m(\Sigma_{V})$. Therefore $\Phi(\sigma_{V})$ is included in the convex hull of $m(\Sigma_{V})$.

The inclusion $m(\Sigma_{V})\subset\Phi(\sigma_{V})$ implies that $co(m(\Sigma_{V}))\subset\Phi(\sigma_{V})$ which proves the equality. ∎

Remark 7**.**

Note that in general the set of extremal points of $\Phi(\sigma_{V})$ is strictly included in $\Phi(\sigma_{V}^{e})=m(\Sigma_{V})$.

Remark 8**.**

If $S^{2r-1}$ denotes the unit sphere in $\mathbb{C}^{r}$ then, since $\Sigma_{V}=\mathcal{V}S^{2r-1}\mathcal{V}^{*}$, we can replace $m(\Sigma_{V})$ with $m(\mathcal{V}S^{2r-1}\mathcal{V}^{*})$ in the previous theorem.

7. On the critical points of the function $F$

The results of the previous section motivates the study of minimum values of $F:\Sigma=\Sigma_{r}\times\Sigma_{s}\to\mathbb{R}_{\geq 0}$ (see (5.5)) attained at extremal points of the sets $\Phi(\sigma_{V})$ and $\Phi(\sigma_{W})$. In this section we describe critical points of $F$ under the assumption that they are attained on pairs of one dimensional projections. This would always be the case if the sets $\Phi(\sigma_{V})$ and $\Phi(\sigma_{W})$ were strictly convex as seen in all the examples we examined where none of the vectors of the standard basis belong to either subspace.

We assume the following:

1. (1)

$(a,b)\in\Sigma$ is a critical point for $F$ 2. (2)

$a$ and $b$ are one dimensional projections in $\mathbb{C}^{r}$ and $\mathbb{C}^{s}$ respectively. 3. (3)

We choose $\tilde{a}\in\mathbb{C}^{r}$ and $\tilde{b}\in\mathbb{C}^{s}$ unit vectors such that

[TABLE] 4. (4)

We denote with

[TABLE]

where $e_{k}$ are the standard base vectors and $\mathcal{V}$, $\mathcal{W}$ some fixed isometries as in (• ‣ 5). 5. (5)

We denote $\alpha_{k}=\langle\tilde{a},x_{k}\rangle$ and $\beta_{k}=\langle\tilde{b},y_{k}\rangle$. Then, using that $\mathcal{V}\tilde{a}=\sum\langle\mathcal{V}\tilde{a},e_{k}\rangle e_{k}$, and that therefore $\tilde{a}=\mathcal{V}^{*}\mathcal{V}\tilde{a}=\sum\langle\mathcal{V}\tilde{a},e_{k}\rangle\mathcal{V}^{*}e_{k}$ we can conclude that $\tilde{a}=\sum\alpha_{k}x_{k}$. Similarly $\tilde{b}=\sum\beta_{k}y_{k}$ can be obtained. 6. (6)

Since $a,b\geq 0$, after some computations follows that the pair $(a,b)$ is a critical point for the function $F$ (see Theorem 6) if and only if

[TABLE]

Observe that $\sum|\alpha_{k}|^{2}=1$. In fact $\alpha_{k}=\langle\mathcal{V}\tilde{a},e_{k}\rangle$ and since $\mathcal{V}$ is an isometry, $||\mathcal{V}\tilde{a}||=1$. Similarly $\sum|\beta_{k}|^{2}=1$.

Now we turn the analysis of equations (7.1). First notice that there exist non trivial complex combinations of the form

[TABLE]

because $r=\dim V<n$ and $s=\dim W<n$.

For each of such pairs, $\xi_{1},\eta_{1};\dots;\xi_{n},\eta_{n}$ consider the system

[TABLE]

obtained from (7.1) identifying each coefficient with the corresponding $\xi_{k}$ and $\eta_{k}$.

Next we multiply the first equation of (7.2) by $\varphi_{k}$ and the second by $\psi_{k}$ (with $|\varphi_{k}|=1$ and $|\psi_{k}|=1$) so that each $\sigma_{k}=\varphi_{k}\xi_{k}$ is real and $\tau_{k}=\psi_{k}\eta_{k}$ is real. Defining $s_{k}=\varphi_{k}\alpha_{k}$ and $t_{k}=\psi_{k}\beta_{k}$ we get from equations (7.2)

[TABLE]

and all the coefficients of these equations are real numbers.

In fact multiplying the first equation in (7.1) by $\tilde{a}$ and the second by $\tilde{b}$ we get

[TABLE]

which shows that $\lambda$ and $\mu$ are real and moreover $\lambda\geq\mu$ because

[TABLE]

In terms of $s_{k}$ and $t_{k}$ equations (7.4) can be rewritten in the form

[TABLE]

The set of equations (7.3) and (7.5) form a complete system of 2n+2 equations with 2n+2 unknowns.

8. Characterization of critical points of $F$

Based on the discussion of the preceding paragraphs we state the following theorem.

Theorem 9**.**

Let $\mathcal{V}:\mathbb{C}^{r}\to\mathbb{C}^{n}$ and $\mathcal{W}:\mathbb{C}^{s}\to\mathbb{C}^{n}$ be fixed isometries such that R$(\mathcal{V})=V\perp$ R$(\mathcal{W})=W$, $\{e_{k}\}_{k=1}^{n}$ be the standard basis of $\mathbb{C}^{n}$ and $a,b$ be unidimensional projections in $\mathbb{C}^{r}$ and $\mathbb{C}^{s}$ respectively. Then the following statements are equivalent,

1. i)

the pair $(a,b)$ is a critical point of the map $F$ (defined in (5.5)), 2. ii)

there exists a pair of unitary vectors $(\tilde{a},\tilde{b})\in\mathbb{C}^{r}\times\mathbb{C}^{s}$ such that $a=\langle\cdot,\tilde{a}\rangle\tilde{a}$ and $b=\langle\cdot,\tilde{b}\rangle\tilde{b}$, and $(\tilde{a},\tilde{b})$ satisfy equations (7.1) for $\alpha_{k}=\langle\tilde{a},\mathcal{V}^{*}e_{k}\rangle$ and $\beta_{k}=\langle\tilde{b},\mathcal{W}^{*}e_{k}\rangle$ for $k=1,\dots,n$, and 3. iii)

there exists a pair of unitary vectors $(\tilde{a},\tilde{b})\in\mathbb{C}^{r}\times\mathbb{C}^{s}$ such that $\tilde{a}=\sum_{k=1}^{n}\overline{\varphi}_{k}s_{k}\mathcal{V}^{*}e_{k}$ and $\tilde{b}=\sum_{k=1}^{n}\overline{\psi}_{k}t_{k}\mathcal{W}^{*}e_{k}$, where

1. (a)

$s_{k},t_{k}\in\mathbb{R}$, for $k=1,\dots,n$ and $\sum_{k=1}^{n}s_{k}^{2}=1=\sum_{k=1}^{n}t_{k}^{2}$, 2. (b)

$\varphi_{k},\psi_{k}\in\mathbb{C}$, $|\varphi_{k}|=|\psi_{k}|=1$, for $k=1,\dots,n$ 3. (c)

$\|\sum_{k=1}^{n}\overline{\varphi}_{k}s_{k}\mathcal{V}^{*}e_{k}\|=1=\|\sum_{k=1}^{n}\overline{\psi}_{k}\ t_{k}\mathcal{W}^{*}e_{k}\|$** 4. (d)

there exists $\sigma_{k},\tau_{k}\in\mathbb{R}_{\geq 0}$ such that \displaystyle{\left\{\begin{array}[]{l}\sum_{k=1}^{n}\overline{\varphi}_{k}\sigma_{k}\mathcal{V}^{*}e_{k}=0\\ \sum_{k=1}^{n}\overline{\psi}_{k}\tau_{k}\mathcal{W}^{*}e_{k}=0,\end{array}\right.} 5. (e)

and $s_{k},t_{k}\in\mathbb{R}$, for $k=1,\dots,n$ are solutions of the systems

[TABLE]

Proof.

The equivalence i) $\Leftrightarrow$ ii) has been discussed in the previous section.

ii) $\Rightarrow$ iii) has also been proved at the end of the previous section.

Let us consider the implication iii) $\Rightarrow$ ii).

If we define $\lambda$ and $\mu$ with \left\{\begin{array}[]{rcl}\lambda/2&=&\sum_{k=1}^{n}(s_{k}^{2}-t_{k}^{2})s_{k}^{2}\\ \mu/2&=&\sum_{k=1}^{n}(t_{k}^{2}-s_{k}^{2})t_{k}^{2}\\ \end{array}\right. then $\lambda$, $\mu$, $s_{k},t_{k}$ (for $k=1,\dots,n$) satisfy (7.5). Moreover, iii) (e) implies that they also satisfy (7.3).

Let us now define $\alpha_{k}=\overline{\varphi}_{k}s_{k}$, $\beta_{k}=\overline{\psi}_{k}t_{k}$, for $k=1,\dots,n$, and observe that the equations iii) (c) $\|\sum_{k=1}^{n}\overline{\varphi}_{k}s_{k}\mathcal{V}^{*}e_{k}\|=1=\|\sum_{k=1}^{n}\overline{\psi}_{k}\ t_{k}\mathcal{W}^{*}e_{k}\|$ are equivalent to $\sum_{k=1}^{n}\overline{\varphi}_{k}s_{k}e_{k}=\sum_{k=1}^{n}\alpha_{k}e_{k}\in V$ and $\sum_{k=1}^{n}\overline{\psi}_{k}t_{k}e_{k}=\sum_{k=1}^{n}\beta_{k}e_{k}\in W$. Then if we define

[TABLE]

follows that

[TABLE]

(since $\sum_{k=1}^{n}\alpha_{k}e_{k}\in V$ and $\sum_{k=1}^{n}\beta_{k}e_{k}\in W$).

Now if we define $\xi_{k}=\overline{\varphi}_{k}\sigma_{k}$ and $\eta_{k}=\overline{\psi}_{k}\tau_{k}$, for $k=1,\dots,n$, then iii)(e) implies that equations (7.2) are satisfied and therefore equations (7.1) are also satisfied with $\alpha_{k}=\langle\tilde{a},\mathcal{V}^{*}e_{k}\rangle$ and $\beta_{k}=\langle\tilde{b},\mathcal{W}^{*}e_{k}\rangle$. Then statement ii) holds. ∎

9. Supports that have neighborhoods of $\mathcal{S}_{(r,s)}$ in $\mathcal{F}_{(r,s)}$

Recall that with $\mathcal{F}_{(r,s)}$ we denote the set of pairs $(V,W)$ of orthogonal subspaces $V$ and $W$ of $\mathbb{C}^{n}$ such that $\dim(V)=r$ and $\dim(W)=s$. See Section 2 for its relation with flag manifolds.

In this section we study the existence of supports $(V,W)\in\mathcal{S}_{(r,s)}$ that belong to an open neighborhood of $\mathcal{F}_{(r,s)}$ formed entirely of supports in $\mathcal{S}_{(r,s)}$.

Remark 9**.**

Note that in general, a support $(V,W)$ of $\mathbb{C}^{n}$ in the flag $\mathcal{F}_{(r,s)}$, is not necessarily an interior point of $\mathcal{F}_{(r,s)}$. Consider for example two orthogonal one dimensional subspaces $V=\text{gen}\{v\}$ and $W=\text{gen}\{w\}$ that form a support in $\mathbb{C}^{n}$ ($n\geq 3$). Then their generators must satisfy $|v_{i}|=|w_{i}|$ for $i=1,\dots,n$. Suppose that $v_{1}\neq 0$, and $v_{1}=\rho e^{i\theta}$, $w_{1}=\rho e^{i\beta}$ with $\rho=|v_{1}|=|w_{1}|$ and $\theta,\beta\in[0,2\pi)$, Then for $\varepsilon>0$ consider small perturbations $v_{\varepsilon}$ and $w_{\varepsilon}$ with their first coordinates $(v_{\varepsilon})_{1}=\rho/(1+\varepsilon)e^{i\theta}$ and $(w_{\varepsilon})_{1}=\rho(1+\varepsilon)e^{i\beta}$ and the rest equal to those of $v$ and $w$. Then $\langle v_{\varepsilon},w_{\varepsilon}\rangle=\langle v,w\rangle=0$ but $|(v_{\varepsilon})_{1}|=\rho/(1+\varepsilon)\neq\rho(1+\varepsilon)=|(w_{\varepsilon})_{1}|$ for $\varepsilon>0$. If we denote with $V_{\varepsilon}$ and $W_{\varepsilon}$ the subspaces generated by $v_{\varepsilon}$ and $w_{\varepsilon}$ respectively, the previous calculations prove that there exist pair of subspaces $(V_{\varepsilon},W_{\varepsilon})$ in the flag $\mathcal{F}_{(1,1)}$ that do not form a support and that they can be chosen as close to $(V,W)$ as desired (taking $\varepsilon\to 0$). Therefore $(V,W)$ is not an interior point of $\mathcal{F}_{(1,1)}$.

Theorem 10**.**

Let $n\in\mathbb{N}$, $n\geq 3$. Then, there exists a support $(V_{n},W_{n})$ in $\mathbb{C}^{n}$ that is an interior point of the flag $\mathcal{F}_{(r,s)}$ for certain $r$, $s$ $<n$.

Proof.

We will use the examples described in the appendices in the cases $n=3$, $n=4$ and $n=5$ where some cases of supports that are interior points of the flags $\mathcal{F}_{(2,1)}$, $\mathcal{F}_{(2,1)}$ and $\mathcal{F}_{(3,1)}$ are shown.

Consider now the supports $(V,W)$ of $\mathbb{C}^{3}$, $\mathbb{C}^{4}$ and $\mathbb{C}^{5}$ described in appendices A, B and C respectively. We will also denote with $V_{3}$, $V_{4}$ and $V_{5}$ the matrices whose columns are defined with the generators of the corresponding subspaces described in each case in the mentioned appendices. $M_{3}=V_{3}\circ\overline{V_{3}}$, $M_{4}=V_{4}\circ\overline{V_{4}}$ and $M_{5}=V_{5}\circ\overline{V_{5}}$ are also the matrices defined there. Similarly $W_{3}$, $W_{4}$ and $W_{5}$ will denote the matrices whose unique column is the generator of the corresponding subspace $W$. In each case these supports are interior points of the corresponding flag manifolds.

Observe that for any $n\in\mathbb{N}$, $n\geq 3$, there exist $h,k,l\in\mathbb{N}$ such that $n=3h+4k+5l$. Let us now fix a triple of those $h,k$ and $l$ and consider the subspaces $V$ and $W$ defined as follows. $V$ is generated by the columns of the following $n\times n$ block matrix $V_{n}$ formed with $h$ copies of $V_{3}$, $k$ of $V_{4}$ and $l$ of $V_{5}$ in the diagonal

[TABLE]

and $W_{n}$ is the $n\times 1$ matrix formed with $h$ copies of $W_{3}$, $k$ of $W_{4}$ and $l$ of $W_{5}$ concatenated (where $W_{i}$, $i=1,2,3$ are the subspaces used in the appendices A, B and C). The transpose of $W_{n}$ is:

[TABLE]

Now consider the subspace $V\subset\mathbb{C}^{n}$, generated by the columns of $V_{n}$ and $W\in\mathbb{C}^{n}$ generated by $W_{n}$. Then it can be verified that $V$ is orthogonal to $W$ and (since $\dim(V_{3})=\dim(V_{4})=2$ and $\dim(V_{5})=3$) that $\dim(V)=2h+2k+3l$ and $\dim(W)=1$. Moreover, considering $M_{n}=V_{n}\circ\overline{V_{n}}\in M_{n}(\mathbb{R}_{+})$ it is easy to see that $\det(M_{n})=\det(M_{3})^{h}\det(M_{4})^{k}\det(M_{5})^{l}\neq 0$ because every factor is non-zero (see A, B and C). Then, the linear system

[TABLE]

has a unique solution $X\in\mathbb{R}^{n\times 1}$. The concrete solution $X$ can be found considering the examples of the Appendix, and satisfies $X_{i,1}>0$ for all $i=1,\dots,n$. Thus the pair $(V,W)\in\mathcal{F}_{(2h+2k+3l,1)}\subset\mathbb{C}^{n}=\mathbb{C}^{3h+4k+5l}$ and is a support as in Definition 1 (consider the vectors $v^{i}=\sqrt{X_{i,1}}v_{i}$ in the conditions (1.4), for $i=1,\dots,n$, and $v_{i}$ the $i^{\text{th}}$ column of $V_{n}$).

Now consider small perturbations $V^{\prime}$ and $W^{\prime}$ of the subspaces $V$ and $W$ such that the dimensions of the perturbed subspaces are conserved and $V^{\prime}\perp W^{\prime}$ holds. That is, the pair $(V^{\prime},W^{\prime})\in\mathcal{F}{(2h+2k+3l,1)}$ and is near $(V,W)$. Then, we can choose $n$ vectors of $V^{\prime}$ close to the ones in the columns of $V_{n}$ such that they generate $V^{\prime}$. Similarly for $W^{\prime}$. Let us denote with $V_{n}^{\prime}$ and $W_{n}^{\prime}$ the matrices such that its columns are the respective generators mentioned. Moreover, $V^{\prime}$ and $W^{\prime}$ can be chosen in a neighborhood of $V$ and $W$ in such a way that the pair of matrices $V_{n}^{\prime}$ and $W_{n}^{\prime}$ satisfy that

1. (1)

$\det(V_{n}^{\prime}\circ\overline{V_{n}^{\prime}})\neq 0$, 2. (2)

the unique solution $X\in\mathbb{C}^{n\times 1}$ of $(V_{n}^{\prime}\circ\overline{V_{n}^{\prime}})X=W_{n}^{\prime}\circ\overline{W_{n}^{\prime}}$ satisfies $X_{i,1}>0$.

This implies that the pair $(V^{\prime},W^{\prime})$ is a support according to Definition 1. Then $(V,W)$ is an interior point of $\mathcal{F}_{(2h+2k+3l,1)}$. ∎

Remark 10**.**

Observe that in the decomposition used in the previous proof given by $n=3h+4k+5l$, with $h,k,l\in\mathbb{N}$, the term $5l$ is only needed for $n=5$. Every $n\in\mathbb{N}\setminus\{1,2,5\}$ can be written as $n=3h+4k$.

Remark 11**.**

Note that at the end of the proof of the previous theorem if the subspaces $V^{\prime}$ and $W^{\prime}$ are not required to be orthogonal they still satisfy the conditions (1) and (2) if they are close enough to $V$ and $W$.

Here we present examples of supports in low dimensions that are interior points of flag manifolds.

Appendix A Example of a support in ${\mathbb{C}}^{3}$ that is an interior point of

$\mathcal{F}_{(2,1)}$

Let us consider the dimension 2 subspace $V\subset\mathbb{C}^{3}$ generated by the following norm one vectors:

[TABLE]

and the subspace $W$ generated by $w=\left(\frac{5-i}{2\sqrt{15}},\left(\frac{1}{2}-\frac{i}{2}\right)\sqrt{\frac{3}{5}},\frac{2}{\sqrt{15}}\right)$, that is orthogonal to $V$.

Then, if $a_{1}=\frac{115667}{303810}$, $a_{2}=\frac{85199}{222794}$ and $a_{3}=\frac{395794}{1670955}$ direct computations show that $\sum_{i=1}^{3}a_{i}=1$ and that if $V_{3}=(v^{1},v^{2},v^{3})$ and $M_{3}=V_{3}\circ\overline{V_{3}}=(v^{1}\circ\overline{v^{1}},v^{2}\circ\overline{v^{2}},v^{3}\circ\overline{v^{3}})$ are the $3\times 3$ matrices with columns $v^{i}$ and respectively $v^{i}\circ\overline{v^{i}}$, for $i=1,2,3$, then

[TABLE]

where $a=(a_{1},a_{2},a_{3})$, and $a^{t}$ is its transpose (a column matrix). This proves that the pair $(V,W)$ is a support (consider the vectors $v^{i}=\sqrt{a_{i}}\,v^{i}$, for $i=1,2,3$ and the definition (1.4)). Moreover, it can be checked that the matrix $M_{3}$ (the one involved in the equation (A.1)) has non-zero determinant (which proves that the numbers $a_{i}$, $i=1,2,3$ are unique).

Then, it can be proved that the support $(V,W)$ is interior in the set of $(2,1)$ flags $\mathcal{F}_{(2,1)}$ of $\mathbb{C}^{3}$. This follows because continuous and small perturbations $w^{\prime}$ and $(v^{\prime})^{i}$ of the vectors $w$ and $v^{i}$ (with the condition $w^{\prime}\perp(v^{\prime})^{i}$, for $i=1,2,3$), produce a non-zero determinant of the perturbed corresponding matrix $M_{3}^{\prime}$ with columns $(v^{\prime})^{i}\circ\overline{(v^{\prime})^{i}}$ for $i=1,2,3$. Then there are unique solutions $a_{i}^{\prime}>0$ of the corresponding equation (A.1) for the new vectors $(v^{\prime})^{i}$ and $w^{\prime}$. This proves that there exists a neighborhood of $(V,W)$ in $\mathcal{F}_{(2,1)}$ such that every pair $(V^{\prime},W^{\prime})$ belonging to it is a support according to Definition 1.

Appendix B Example of a support in $\mathbb{C}^{4}$ that is an interior point of

$\mathcal{F}_{(2,1)}$

Let $V\subset\mathbb{C}^{4}$ be the subspace of dimension 2 generated by the following norm one vectors:

[TABLE]

and $W$ the subspace generated by $w=\left(\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{2}},\frac{\frac{1}{2}-\frac{i}{2}}{\sqrt{2}},\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{2}},\frac{\frac{1}{2}+\frac{i}{2}}{\sqrt{2}}\right)$.

If $a_{1}=\frac{20559837596768881}{124590980225106843}$, $a_{2}=\frac{96813856451303497}{415303267417022810}$, $a_{3}=\frac{1154873210442508}{8612279739062685}$ and $a_{4}=\frac{49954131355895969}{106792268764377294}$, it can be verified that if $V_{4}=(v^{1},v^{2},v^{3},v^{4})$, $M_{4}=V_{4}\circ\overline{V_{4}}=(v^{1}\circ v^{1},v^{2}\circ v^{2},v^{3}\circ v^{3},v^{4}\circ v^{4})$ and $a=(a_{1},a_{2},a_{3},a_{4})$, then

[TABLE]

The determinant of the matrix $M_{4}$ is non-zero and therefore similar considerations as those made in the previous example in A can be used in order to prove that $(V,W)$ is a support of $\mathbb{C}^{4}$ that is included in an open subset of the flags $\mathcal{F}_{(2,1)}$.

Appendix C Example of a support in $\mathbb{C}^{5}$ that is an interior point of

$\mathcal{F}_{(3,1)}$

Let $V\subset\mathbb{C}^{5}$ be the subspace of dimension 3 generated by the rows of the matrix M_{V}=\left(\begin{array}[]{ccccc}-\frac{19}{50}-\frac{i}{50}&-\frac{2}{25}+\frac{19i}{50}&-\frac{7}{25}+\frac{3i}{25}&\frac{8}{25}+\frac{3i}{25}&\frac{8}{25}+\frac{11i}{50}\\ -\frac{1}{5}+\frac{11i}{50}&\frac{1}{10}+\frac{3i}{25}&\frac{19}{50}-\frac{i}{5}&\frac{19}{50}+\frac{3i}{10}&-\frac{21}{50}\\ \frac{29}{50}&-\frac{1}{50}+\frac{3i}{10}&-\frac{1}{10}-\frac{8i}{25}&\frac{1}{5}+\frac{9i}{50}&\frac{1}{5}-\frac{21i}{50}\\ \end{array}\right) and $W$ the subspace generated by $w=\left(\frac{1-i}{\sqrt{10}},\frac{1-i}{\sqrt{10}},\frac{1+i}{\sqrt{10}},\frac{1+i}{\sqrt{10}},\frac{1+i}{\sqrt{10}}\right).$

Now define the coefficient matrix $C=\begin{pmatrix}-\frac{16}{5}+\frac{53i}{10}&-\frac{4}{5}-\frac{5i}{2}&\frac{8}{5}+\frac{29i}{10}\\ 1+\frac{i}{2}&-\frac{3}{5}&\frac{2}{5}+\frac{13i}{10}\\ -\frac{13i}{5}&-\frac{11}{5}+\frac{29i}{10}&\frac{4}{5}+i\\ 1+\frac{3i}{5}&\frac{3}{5}+\frac{7i}{5}&\frac{1}{2}+\frac{7i}{10}\\ \frac{13}{10}+\frac{3i}{2}&-\frac{2}{5}+\frac{9i}{5}&-\frac{14}{5}-\frac{3i}{10}\end{pmatrix}$, and consider the 5 vectors belonging to $V$ obtained from the rows of the product $C\cdot M_{V}\in\mathbb{C}^{5\times 5}$. Let us denote those vectors (rows) with $v^{1}$, $v^{2}$, $v^{3}$, $v^{4}$ and $v^{5}$. Then it can be checked that for $a_{1}=\frac{38245034600180718292066117}{93505493283505350729949090}$, $a_{2}=\frac{876893808432404350620802}{9350549328350535072994909}$, $a_{3}=\frac{11840789324853298629489761}{93505493283505350729949090}$, $a_{4}=\frac{11483749488079211997737796}{46752746641752675364974545}$ and $a_{5}=\frac{1168323229798886630670960}{9350549328350535072994909}$ the equality $\sum_{i=1}^{5}a_{i}=1$ holds and if $V_{5}=(v^{1},v^{2},v^{3},v^{4},v^{5})$, $M_{5}=V_{5}\circ\overline{V_{5}}=(v^{1}\circ v^{1},v^{2}\circ v^{2},v^{3}\circ v^{3},v^{4}\circ v^{4},v^{5}\circ v^{5})$ and $a=(a_{1},a_{2},a_{3},a_{4},a_{5})$, then

[TABLE]

The determinant of the matrix $M_{5}$ involved in equation (C.1) is non-zero and therefore similar considerations as those made in the previous examples of the Appendix can be made in order to prove that $(V,W)$ is a support that is included in an open subset of the flags $\mathcal{F}_{(3,1)}$ of $\mathbb{C}^{5}$.

Remark 12**.**

Note that the steps used to prove that the previous example $(V_{5},W_{5})$ is an interior point of $\mathcal{F}_{(3,1)}$ in $\mathbb{C}^{5}$ cannot be followed if the dimensions of the subspaces were $\dim(V)=2$ and $\dim(W)=1$ as in A and B. This is because if $\dim(V)=2$ then $\operatorname{rank}{M_{5}}=\operatorname{rank}(V_{5}\circ\overline{V_{5}})\leq\operatorname{rank}(V_{5})\operatorname{rank}(\overline{V_{5}})=4$, and therefore $\det(M_{5})=0$ in this case (for any choice of $V_{5}$). This is not enough to asseverate that there is not a support in $\mathbb{C}^{5}$ that is an interior point of $\mathcal{F}_{(2,1)}$, but we have not found an example with these dimensions.