Orthogonality to matrix subspaces, and a distance formula

TL;DR

This paper provides a precise condition for matrix orthogonality to subspaces and derives a formula for the distance from matrices to certain subalgebras, advancing understanding of matrix geometry.

Contribution

It introduces a necessary and sufficient condition for matrix orthogonality to subspaces and a new distance formula to unital C*-subalgebras.

Findings

Characterization of Birkhoff-James orthogonality to subspaces

Explicit distance formula to unital C*-subalgebras

Enhanced understanding of matrix subspace geometry

Abstract

We obtain a necessary and sufficient condition for a matrix $A$ to be Birkhoff-James orthogonal to any subspace $\mathscr W$ of $\mathbb M_n(\mathbb C)$. Using this we obtain an expression for the distance of $A$ from any unital $C^*$ subalgebra of $\mathbb M_n(\mathbb C)$.

Equations

$$\Vert A\Vert =\underset{x\in {\mathbb{C}}^n,\Vert x\Vert =1}{\max }\Vert Ax\Vert$$

$$\Vert A+W\Vert \ge \Vert A\Vert \text{ for all }W\in \mathcal{W}.$$

$$\mathrm{dist}(A,\mathcal{W})=\min \left\{\Vert A-W\Vert :W\in \mathcal{W}\right\}.$$

$$\mathrm{dist}(A,\mathbb{C}I{)}^2=\max \left\{\mathrm{tr}(A^{\ast }AP)-|\mathrm{tr}(AP){|}^2:P\ge 0,\mathrm{tr}P=1\right\}.$$

$$\Phi (A^{\ast }A)-\Phi (A{)}^{\ast }\Phi (A)\le \mathrm{dist}(A,\mathbb{C}I{)}^2.$$

$${\mathcal{C}}_{\mathcal{B}}(BX)=B{\mathcal{C}}_{\mathcal{B}}(X)\text{ and }{\mathcal{C}}_{\mathcal{B}}(XB)={\mathcal{C}}_{\mathcal{B}}(X)B\text{ for all }B\in \mathcal{B},X\in {\mathbb{M}}_n(\mathbb{C}).$$

$$\mathcal{C}(X)=\left[\begin{array}[]{cccc}X_{11}&&&\\ &X_{22}&&\\ &&\ddots&\\ &&&X_{kk}\end{array}\right].$$

$$\mathrm{dist}(A,\mathcal{B}{)}^2=\max \left\{\mathrm{tr}\left(A^{\ast }AP-{\mathcal{C}}_{\mathcal{B}}(AP{)}^{\ast }{\mathcal{C}}_{\mathcal{B}}(AP){\mathcal{C}}_{\mathcal{B}}(P{)}^{-1}\right):P\ge 0,\mathrm{tr}P=1\right\},$$

$$f(y)-f(x)\ge \mathrm{Re}{v}^{\ast }(y-x)\text{ for all }y\in X.$$

$$\partial (g\circ L)(x)=S^{\ast }\partial g(L(x)),$$

$$⟨S^{\ast }(y),x⟩=⟨y,S(x)⟩\text{ for all }x\in X\text{ and }y\in Y.$$

$$\partial \Vert A\Vert =\mathrm{conv}\{u{v}^{\ast }:\Vert u\Vert =\Vert v\Vert =1,Av=\Vert A\Vert u\},$$

$$\Vert A+W{\Vert }^2$$

$$\Vert T\Vert =\underset{\Vert X{\Vert }_1=1}{\sup }|\mathrm{tr}(TX)|,$$

$$\Vert A+W{\Vert }^2$$

$$\Vert A+W\Vert \ge \Vert A\Vert \text{ for all }W\in \mathcal{W}.$$

$$L(W)=A+S(W).$$

$$(g\circ L)(W)\ge (g\circ L)(0),$$

$$0\in S^{\ast }\partial \Vert A\Vert .$$

$$S^{\ast }\partial \Vert A\Vert =\mathrm{conv}\left\{{\mathrm{S}}^{\ast }(\mathrm{u}{\mathrm{v}}^{\ast }):\Vert u\Vert =\Vert v\Vert =1,\mathrm{Av}=\Vert A\Vert u\right\}.$$

$$S^{\ast }\left(\sum t_i{u}_{(i)}{v}_{(i)}^{\ast }\right)=0.$$

$$AP$$

$$A^{\ast }AP$$

$$(s_j^2-\Vert A{\Vert }^2)p_j=0\text{ for all }k+1\le j\le n.$$

$$\mathrm{dist}(A,\mathcal{B})=\underset{W\in \mathcal{B}}{\min }\Vert A-W\Vert .$$

$$\mathrm{dist}(A,\mathcal{B})=\mathrm{dist}(\tilde{A},{\oplus }_i{\mathbb{M}}_{n_i}(\mathbb{C})).$$

$$\max \left\{\mathrm{tr}\left(A^{\ast }AP-{\mathcal{C}}_{\mathcal{B}}(AP{)}^{\ast }{\mathcal{C}}_{\mathcal{B}}(AP){\mathcal{C}}_{\mathcal{B}}(P{)}^{-1}\right):P\ge 0,\mathrm{tr}P=1\right\}$$

$$=\max \left\{\mathrm{tr}\left({\tilde{A}}^{\ast }\tilde{A}\tilde{P}-\mathcal{C}(\tilde{A}\tilde{P}{)}^{\ast }\mathcal{C}(\tilde{A}\tilde{P})\mathcal{C}(\tilde{P}{)}^{-1}\right):\tilde{P}\ge 0,\mathrm{tr}\tilde{P}=1\right\},$$

$$\mathrm{tr}(XY)=\mathrm{tr}(YX),$$

$$\mathrm{tr}\left(A^{\ast }AP-\mathcal{C}(AP{)}^{\ast }\mathcal{C}(AP)\mathcal{C}(P{)}^{-1}\right)=\mathrm{tr}\left(V^{\ast }{A}^{\ast }APV-V^{\ast }\mathcal{C}(AP{)}^{\ast }\mathcal{C}(AP)\mathcal{C}(P{)}^{-1}V\right).$$

$$\mathrm{tr}\left({\tilde{A}}^{\ast }\tilde{A}\tilde{P}-\mathcal{C}(\tilde{A}\tilde{P}{)}^{\ast }\mathcal{C}(\tilde{A}\tilde{P})\mathcal{C}(\tilde{P}{)}^{-1}\right).$$

Full text

*AMS classification: * 15A60, 15A09, 47A12

*Keywords: * Birkhoff-James orthogonality, Subdifferential, Singular value decomposition, Moore-Penrose inverse, Pinching, Variance.

1 Introduction

Let $\mathbb{M}_{n}(\mathbb{C})$ be the space of $n\times n$ complex matrices and let $\mathscr{W}$ be any subspace of $\mathbb{M}_{n}(\mathbb{C})$. For any $A\in\mathbb{M}_{n}(\mathbb{C})$, let

[TABLE]

be the operator norm of $A$. Then $A$ is said to be (Birkhoff-James) orthogonal to $\mathscr{W}$ if

[TABLE]

The space $\mathbb{M}_{n}(\mathbb{C})$ is a complex Hilbert space under the inner product $\langle A,B\rangle_{c}={\operatorname{tr\ }}(A^{*}B)$ and a real Hilbert space under the inner product $\langle A,B\rangle_{r}={\operatorname{Re}}\ {\operatorname{tr\ }}(A^{*}B)$. Let $\mathscr{W}^{\perp}$ be the orthogonal complement of $\mathscr{W}$, where the orthogonal complement is with respect to the usual Hilbert space orthogonality in $\mathbb{M}_{n}(\mathbb{C})$ with the inner product $\langle\cdot,\cdot\rangle_{c}$ or $\langle\cdot,\cdot\rangle_{r}$, depending upon whether $\mathscr{W}$ is a real or complex subspace. Note that if $A\in\mathscr{W}^{\perp}$ such that ${\operatorname{tr\ }}(A^{*}A)=\|A\|^{2}$, then $A$ is orthogonal to $\mathscr{W}$.

Bhatia and $\check{\text{S}}$emrl [6] obtained an interesting characterisation of orthogonality when $\mathscr{W}=\mathbb{C}B$, where $B$ is any matrix in $\mathbb{M}_{n}(\mathbb{C})$. They showed that $A$ is orthogonal to $\mathbb{C}B$ if and only if there exists a unit vector $x$ such that $\|Ax\|=\|A\|$ and $\langle Ax,Bx\rangle=0$. In other words, $A$ is orthogonal to $\mathbb{C}B$ if and only if there exists a positive semidefinite matrix $P$ of rank one such that ${\operatorname{tr\ }}P=1,\ {\operatorname{tr\ }}A^{*}AP=\|A\|^{2}$ and $AP\in(\mathbb{C}B)^{\perp}.$ Such positive semidefinite matrices with trace 1 are called density matrices. We use the notation $P\geq 0$ to mean $P$ is positive semidefinite.

Let $\mathscr{W}=\mathbb{D}_{n}(\mathbb{R}),$ the subspace of all diagonal matrices with real entries, and let $A$ be any Hermitian matrix. Then $A$ is called minimal if $\|A+D\|\geq\|A\|$ for all $D\in\mathbb{D}_{n}(\mathbb{R})$. Andruchow, Larotonda, Recht, and Varela [1, Theorem 1] showed that a Hermitian matrix $A$ is minimal if and only if there exists a density matrix $P$ such that $\ PA^{2}=\|A\|^{2}P$ and all diagonal entries of $PA$ are zero. In our notation, $A$ is minimal is same as saying that $A$ is orthogonal to the subspace $\mathbb{D}_{n}(\mathbb{R})$. If $A$ is Hermitian, then note that $A$ is orthogonal to $\mathbb{D}_{n}(\mathbb{R})$ if and only if $A$ is orthogonal to $\mathbb{D}_{n}(\mathbb{C})$. Now $\mathbb{D}_{n}(\mathbb{C})^{\perp}$ is the subspace of all matries such that their diagonal entries are zero. The condition $\ PA^{2}=\|A\|^{2}P$ is same as $A^{2}P=\|A\|^{2}P$ and diagonal entries of $PA$ are same as diagonal entries of $AP$. Therefore Theorem 1 in [1] can be interpreted as follows. A Hermitian matrix $A$ is orthogonal to $\mathbb{D}_{n}(\mathbb{C})$ if any only if $A^{2}P=\ |A\|^{2}P$ and $AP\in\mathbb{D}_{n}(\mathbb{C})^{\perp}$. The following theorem is a generalization of this result as well as Bhatia-$\check{\text{S}}$emrl theorem.

Theorem 1**.**

Let $A\in\mathbb{M}_{n}(\mathbb{C})$ and let $m(A)$ be the multiplicity of the maximum singular value $\|A\|$ of $A$. Let $\mathscr{W}$ be any (real or complex) subspace of $\mathbb{M}_{n}(\mathbb{C}).$ Then $A$ is orthogonal to $\mathscr{W}$ if and only if there exists a density matrix $P$ of complex rank at most $m(A)$ such that $A^{*}AP=\|A\|^{2}P$ and $AP\in\mathscr{W}^{\perp}$. (If rank $P=\ell$, then $P$ has the form $P=\displaystyle\sum_{i=1}^{\ell}t_{i}v_{(i)}v_{(i)}^{*}$ where $v_{(i)}$ are unit vectors such that $A^{*}Av_{(i)}=\|A\|^{2}v_{(i)}$ and $t_{i}$ are such that $0\leq t_{i}\leq 1$ and $\displaystyle\sum_{i=1}^{\ell}t_{i}=1$.)

Here, $m(A)$ is the best possible upper bound on rank $P$. This has been illustrated later in Remark 4 in Section 4. When $\mathscr{W}=\mathbb{C}B$, the above theorem says that $A$ is orthogonal to $\mathbb{C}B$ if and only if there exists a $P\geq 0$ of the form $P=\displaystyle{\sum_{i=1}^{\ell}}t_{i}v_{(i)}v_{(i)}^{*}$ such that $\|v_{(i)}\|=1$, $A^{*}Av_{(i)}=\|A\|^{2}v_{(i)}$ and $\displaystyle\sum_{i=1}^{\ell}t_{i}\langle B^{*}Av_{(i)},v_{(i)}\rangle=0$. By the Hausdorff-Toeplitz theorem, we get a unit vector $v$ such that $A^{*}Av=\|A\|^{2}v$ and $\langle B^{*}Av,v\rangle=0$. The first condition is stronger than that in [6, Theorem 1.1].

Let ${\operatorname{dist}}(A,\mathscr{W})$ denote the distance of a matrix $A$ from the subspace $\mathscr{W}$, defined as

[TABLE]

Audenaert [2] showed that when $\mathscr{W}=\mathbb{C}I$, then

[TABLE]

Further the maximisation over $P$ on the right hand side of (2) can be restricted to density matrices of rank 1. The quantity ${\operatorname{tr\ }}(A^{*}AP)-|{\operatorname{tr\ }}(AP)|^{2}$ is called the variance of $A$ with respect to the density matrix $P$. Bhatia and Sharma [7] showed that if $\Phi:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{k}(\mathbb{C})$ is any positive unital linear map, then

[TABLE]

By choosing $\Phi(A)={\operatorname{tr\ }}(AP)$ for different density matrices $P$, they obtained various interesting bounds on ${\operatorname{dist}}(A,\mathbb{C}I)^{2}.$

It would be interesting to have a generalisation of (2) with $\mathbb{C}I$ replaced by any unital $C^{*}$ subalgebra of $\mathbb{M}_{n}(\mathbb{C})$. (This problem has also been raised by M. Rieffel in [13].) Let $\mathscr{B}$ be any unital $C^{*}$ subalgebra of $\mathbb{M}_{n}(\mathbb{C})$. Let $\mathcal{C}_{\mathscr{B}}:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathscr{B}$ denote the projection of $\mathbb{M}_{n}(\mathbb{C})$ onto $\mathscr{B}$. We note that $\mathcal{C}_{\mathscr{B}}$ is a bimodule map:

[TABLE]

In particular, when $\mathscr{B}$ is the subalgebra of block diagonal matrices, the matrix $\mathcal{C}_{\mathscr{B}}(X)$ is called a pinching of $X$ and is denoted by $\mathcal{C}(X)$. It is defined as follows. If X=\left[\begin{array}[]{cccc}X_{11}&\cdots&X_{1k}\\ X_{21}&\cdots&X_{2k}\\ \vdots&\vdots&\vdots\\ X_{k1}&\cdots&X_{kk}\end{array}\right] then

[TABLE]

Properties of pinchings are studied in detail in [3] and [4].

Our next result provides a generalisation of (2) for distance of $A$ to any unital $C^{*}$ subalgebra of $\mathbb{M}_{n}(\mathbb{C})$.

Theorem 2**.**

Let $\mathscr{B}$ be any unital $C^{*}$ subalgebra of $\mathbb{M}_{n}(\mathbb{C})$. Let $\mathcal{C}_{\mathscr{B}}:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathscr{B}$ denote the projection of $\mathbb{M}_{n}(\mathbb{C})$ onto $\mathscr{B}$.

Then

[TABLE]

where $\mathcal{C}_{\mathscr{B}}(P)^{-1}$ denotes the Moore-Penrose inverse of $\mathcal{C}_{\mathscr{B}}(P)$. The maximum on the right hand side of (5) can be restricted to rank $P\leq m(A)$.

We prove Theorem 1 using ideas of subdifferential calculus. A brief summary of these is given in Section 2. The proofs are given in Section 3.

2 Preliminaries

Let $X$ be a complex Hilbert space. Let $f:X\rightarrow\mathbb{R}$ be a convex function. Then the subdifferential of $f$ at any point $x\in X$, denoted by $\partial f(x)$, is the set of $v^{*}\in X^{*}$ such that

[TABLE]

It follows from (6) that $f$ is minimized at $x$ if and only if $0\in\partial f(x)$.

We use an idea similar to the one in [8, Theorem 2.1]. Let $f(W)=\|A+W\|$. This is the composition of two functions namely $W\rightarrow A+W$ from $\mathscr{W}$ into $\mathbb{M}_{n}(\mathbb{C})$ and $T\rightarrow\|T\|$ from $\mathbb{M}_{n}(\mathbb{C})$ into $\mathbb{R}_{+}$. Thus we need to find subdifferentials of composition maps. For that we need a chain rule.

Proposition 1**.**

Let $X,Y$ be any two Hilbert spaces. Let $g:Y\rightarrow\mathbb{R}$ be a convex function. Let $S:X\rightarrow Y$ be a linear map and let $L:X\rightarrow Y$ be the affine map defined by $L(x)=S(x)+y_{0}$, for some $y_{0}\in Y$. Then

[TABLE]

where $S^{*}$ is the adjoint of $S$ defined as

[TABLE]

In our setting, $g$ is the map $T\rightarrow\|T\|$. The subdifferential of this map has been calculated by Watson [14].

Proposition 2**.**

Let $A\in\mathbb{M}_{n}(\mathbb{C})$. Then

[TABLE]

where $\mathop{{\rm conv}}D$ denotes the convex hull of a set $D$.

These elementary facts can be found in [11]. In this book the author deals with convex functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}$. The same proofs can be extended to functions $f:X\rightarrow\mathbb{R}$, where $X$ is any Hilbert space.

3 Proofs

Proof of Theorem 1  Suppose there exists a positive semidefinite $P$ with ${\operatorname{tr\ }}P=1$ such that $A^{*}AP=\|A\|^{2}P$ and $AP\in\mathscr{W}^{\perp}$. Then for any $W\in\mathscr{W}$

[TABLE]

Now for any $T\in\mathbb{M}_{n}(\mathbb{C})$,

[TABLE]

where $\|\cdot\|_{1}$ denotes the trace norm. So,

[TABLE]

Since $AP\in\mathscr{W}^{\perp}$, we have ${\operatorname{Re}}\ {\operatorname{tr\ }}(A^{*}WP)={\operatorname{Re}}\ {\operatorname{tr\ }}(W^{*}AP)=0$. The matrices $W^{*}W$ and $P$ are positive semidefinite, therefore ${\operatorname{tr\ }}(W^{*}WP)\geq 0$ and by our assumption, ${\operatorname{tr\ }}(A^{*}AP)=\|A\|^{2}$. Using these in (10) we get that $\|A+W\|^{2}\geq\|A\|^{2}$.

Conversely, suppose

[TABLE]

Let $S:\mathscr{W}\rightarrow\mathbb{M}_{n}(\mathbb{C})$ be the inclusion map. Then $S^{*}:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathscr{W}$ is the projection onto the subspace $\mathscr{W}$. Let $L:\mathscr{W}\rightarrow\mathbb{M}_{n}(\mathbb{C})$ be the map defined as

[TABLE]

Let $g:\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{R}$ be the map taking an $n\times n$ matrix $W$ to $\|W\|$. Then (11) can be rewritten as

[TABLE]

that is, $g\circ L$ is minimized at [math]. Therefore $0\in\partial(g\circ L)(0)$. Using Proposition 1, we get

[TABLE]

By Proposition 2,

[TABLE]

From (12) and (13) it follows that there exist unit vectors $u_{(i)},v_{(i)}$ such that $Av_{(i)}=\|A\|u_{(i)}$ and numbers $t_{i}$ such that $0\leq t_{i}\leq 1$, $\sum t_{i}=1$ and

[TABLE]

Let $P=\sum t_{i}v_{(i)}v_{(i)}^{*}$. Then $P\geq 0$ and ${\operatorname{tr\ }}P=1$. Note that

[TABLE]

So, from (14) we get $S^{*}(AP)=0$, that is, $AP\in\mathscr{W}^{\perp}$. Since each $v_{(i)}$ is a right singular vector for $A$, we have $A^{*}Av_{(i)}=\|A\|^{2}v_{(i)}$. Using this we obtain

[TABLE]

Now let $m(A)=k$. We now show that if $P$ satisfies (15), then rank $P\leq k$. First note that $A^{*}A$ and $P$ commute and therefore can be diagonalised simultaneously. So we can assume $A^{*}A$ and $P$ in (15) to be diagonal matrices. By hypothesis $k$ of the diagonal entries of $A^{*}A$ are equal to $\|A\|^{2}$. Let A^{*}A=\left[\begin{array}[]{ccccccc}\|A\|^{2}&&&&&&\\ &\ddots&&&&&\\ &&\|A\|^{2}&&&\\ &&&s_{k+1}^{2}&&\\ &&&&\ddots&\\ &&&&&s_{n}^{2}\end{array}\right], where $s_{j}<\|A\|$ for all $k+1\leq j\leq n$. If P=\left[\begin{array}[]{ccc}p_{1}&&\\ &\ddots&\\ &&p_{n}\end{array}\right], then from (15) we obtain

[TABLE]

So $p_{j}=0$ for all $k+1\leq j\leq n$. Hence rank $P\leq k$. ∎

Proof of Theorem 2  We first show that it is sufficient to prove the result when $\mathscr{B}$ is a subalgebra of block diagonal matrices in $\mathbb{M}_{n}(\mathbb{C})$. If $\mathscr{B}$ is any subalgebra of $\mathbb{M}_{n}(\mathbb{C})$ then there exist $n_{1},n_{2},\ldots,n_{k}$ with $\sum_{i}n_{i}=n$ such that $\mathscr{B}$ is $*$-isomorphic to $\oplus_{i}\mathbb{M}_{n_{i}}(\mathbb{C})$, the $*$-isomorphism $\varphi:\mathscr{B}\rightarrow\oplus_{i}\mathbb{M}_{n_{i}}(\mathbb{C})$ being $\varphi(X)=V^{*}XV$ for some unitary matrix $V\in\mathbb{M}_{n}(\mathbb{C})$ (see [9, p. 249], [10, p. 74]). By definition

[TABLE]

Let $\tilde{A}$ denote the matrix $V^{*}AV$. Since $\|\cdot\|$ is unitarily invariant, we get

[TABLE]

Next we show that for any density matrix $P$,

[TABLE]

where $\mathcal{C}$ is the pinching map as defined in (4). Since

[TABLE]

we have

[TABLE]

Now note that for any $X\in\mathbb{M}_{n}(\mathbb{C}),$ $V^{*}\mathcal{C}(X)V=\mathcal{C}(V^{*}XV)$. . Therefore the above expression is same as

[TABLE]

This gives (17). So it is enough to prove (5) when $\mathscr{B}$ is a subalgebra of block diagonal matrices. We first show that

[TABLE]

Let $P$ be any density matrix. Then ${\operatorname{tr\ }}(A^{*}AP)\leq\|A\|^{2}.$ Therefore

[TABLE]

Let $B\in\mathscr{B}$. Applying the translation $A\rightarrow A+B$ in (21) we get

[TABLE]

We show that the expression on the left hand side is invariant under this translation. By expanding the expression on the left hand side of (22), we get

[TABLE]

We show that except for the first term, $\left({\operatorname{tr\ }}\left(A^{*}AP-\mathcal{C}(AP)^{*}\ \mathcal{C}(AP)\ \mathcal{C}(P)^{-1}\right)\right)$, the rest of the terms in (23) are zero. We shall prove that the second term

[TABLE]

in (23) is zero. The proof for the other two terms is similar.

By using (3), the expression in (24) is equal to

[TABLE]

By (18) this is equal to

[TABLE]

If $\mathcal{C}(P)$ is invertible then this is clearly zero. So let $\mathcal{C}(P)$ be not invertible. This means that if \mathcal{C}(P)=\left[\begin{array}[]{ccc}P_{1}&&\\ &\ddots&\\ &&P_{k}\end{array}\right], then there exists $i,1\leq i\leq k,$ such that $P_{i}$ is not invertible. Let $U$ denote the block diagonal unitary matrix

[TABLE]

where $U_{i}=I,$ if $P_{i}$ is invertible and U_{i}^{*}P_{i}U_{i}=\left[\begin{array}[]{ccc}\Lambda_{i}&\\ &O\\ \end{array}\right], if $P_{i}$ is not invertible. (Here $\Lambda_{i}$ is the diagonal matrix with eigenvalues of $P_{i}$ as its diagonal entries.) Let $X^{\prime}$ denote the matrix $U^{*}XU$. Then from (3) and (18), we get that the expression in (25) is same as

[TABLE]

Now \mathcal{C}(P^{\prime})=\left[\begin{array}[]{ccccc}\Lambda_{1}&&&&\\ &O&&&\\ &&\Lambda_{2}&&\\ &&&O&\\ &&&&\ddots\end{array}\right]. Write $A^{\prime}$ and $P^{\prime}$ as $2k$-block matrices, $A^{\prime}=(A^{\prime}_{rs})_{r,s=1,\ldots,2k}\text{ and }P^{\prime}=(P^{\prime}_{rs})_{r,s=1,\ldots,2k},$ respectively such that whenever $P_{i}$ is not invertible, we have $P_{2i-1,2i-1}^{\prime}=\Lambda_{i}$ and $P_{2i,2i}^{\prime}=O$.

The $(r,r)$-entry of $A^{\prime}P^{\prime}$ is $\displaystyle\sum_{s=1}^{2k}A^{\prime}_{rs}P^{\prime}_{sr}$. Suppose $P^{\prime}_{rr}=O$. Since $P^{\prime}\geq 0$, we have $P^{\prime}_{rs}=P^{\prime}_{sr}=O$ for all $s=1,\ldots,2k$. Hence the $(r,r)$-entry of $A^{\prime}P^{\prime}$ is zero. So let $P^{\prime}_{rr}\neq O$. Then the $(r,r)$-entry of $\left(I-\mathcal{C}(P^{\prime})^{-1}\mathcal{C}(P^{\prime})\right)$ is zero. Therefore the expression in (27) is zero, and hence the expression in (25) is zero. Therefore from (22), we obtain

[TABLE]

for all $B\in\mathscr{B}$ and for all density matrices $P$. Equation (20) now follows from here.

To show equality in (20), let ${\operatorname{dist}}(A,\mathscr{B})=\|A_{0}\|,$ where $A_{0}=A-B_{0}$ for some $B_{0}\in\mathscr{B}$. Then $A_{0}$ is orthogonal to $\mathscr{B}$. By Theorem 1 there exists a density matrix $P$ such that

[TABLE]

and

[TABLE]

From (28) we get that

[TABLE]

By using (3), we obtain

[TABLE]

Substituting (29) in (30) we get

[TABLE]

Now consider ${\operatorname{tr\ }}\left(\mathcal{C}(AP)^{*}\ \mathcal{C}(AP)\ \mathcal{C}(P)^{-1}\right)$. From (29) we see that this is same as ${\operatorname{tr\ }}\left(B_{0}^{*}B_{0}\ \mathcal{C}(P)\mathcal{C}(P)^{-1}\mathcal{C}(P)\right)$. If $\mathcal{C}(P)$ is invertible, then this is equal to ${\operatorname{tr\ }}(B_{0}^{*}B_{0}P)$. If $\mathcal{C}(P)$ is not invertible, then we define $U$ as done in (26). From (3) and (18), we obtain

[TABLE]

By definition of $U$, this is equal to ${\operatorname{tr\ }}\left(B_{0}^{\prime*}\ B^{\prime}_{0}\ \mathcal{C}(P^{\prime})\right)$, which again by (3) and (18), is same as ${\operatorname{tr\ }}\left(B_{0}^{*}\ B_{0}\ \mathcal{C}(P)\right)$. Therefore from (31) we have

[TABLE]

4 Remarks

It is clear from the proof of Theorem 1 that the condition $A^{*}AP=\|A\|^{2}P$ can be replaced by the weaker condition ${\operatorname{tr\ }}(A^{*}AP)=\|A\|^{2}$ in the statement of Theorem 1. 2. 2.

As one would expect, the set $\{A:\|A+W\|\geq\|A\|\text{ for all }W\in\mathscr{W}\}$ need not be a subspace. As an example consider the subspace $\mathscr{W}=\mathbb{C}I$ of $\mathbb{M}_{3}(\mathbb{C})$. Let A_{1}=\left[\begin{array}[]{ccc}0&1&0\\ 1&0&1\\ 0&1&0\end{array}\right] and A_{2}=\left[\begin{array}[]{ccc}0&0&1\\ 0&0&0\\ 1&0&0\end{array}\right]. It can be checked from Theorem 1 that $A_{1},A_{2}$ are orthogonal to $\mathscr{W}$. (Take P=\left[\begin{array}[]{ccc}0&0&0\\ 0&1&0\\ 0&0&0\end{array}\right] for $A_{1}$ and P=\left[\begin{array}[]{ccc}1&0&0\\ 0&0&0\\ 0&0&0\end{array}\right] for $A_{2}$, respectively.) Then A_{1}+A_{2}=\left[\begin{array}[]{ccc}0&1&1\\ 1&0&1\\ 1&1&0\end{array}\right], and $\|A_{1}+A_{2}\|=2$. But $\left\|A_{1}+A_{2}-\frac{1}{2}I\right\|=\frac{3}{2}<\|A_{1}+A_{2}\|$. Hence $A_{1}+A_{2}$ is not orthogonal to $\mathscr{W}$. 3. 3.

Let $\mathscr{W}=\{X:{\operatorname{tr\ }}X=0\}$. Then $\mathscr{W}^{\perp}=\mathbb{C}I$. In Section 1, we stated that if $A\in\mathscr{W}^{\perp}$ such that ${\operatorname{tr\ }}(A^{*}A)=\|A\|^{2}$ then $A$ is orthogonal to $\mathscr{W}$. Therefore all the scalar matrices are orthogonal to $\mathscr{W}$. We show that if $A\notin\mathbb{C}I$ then there exists a matrix $W$ with ${\operatorname{tr\ }}W=0$ such that $\|A+W\|<\|A\|$. Let $\mathcal{D}A$ and $\mathcal{O}A$ denote the diagonal and off-diagonal parts of $A$, respectively. Then $\mathcal{O}A\in\mathscr{W}$, $A-\mathcal{O}A=\mathcal{D}A$ and $\|\mathcal{D}A\|\leq\|A\|$. So it is enough to find $W\in\mathscr{W}$ such that $\|\mathcal{D}A+W\|<\|\mathcal{D}A\|$. Let $\mathcal{D}A={\operatorname{diag}}\left(a_{1},\ldots,a_{1},a_{2},\ldots,a_{2},\ldots,a_{k},\ldots,a_{k}\right),$ where each $a_{j}$ occurs on the diagonal $n_{j}$ times and $n_{1}+\cdots+n_{k}=n$. Assume $\|\mathcal{D}A\|=1$. Take $W={\operatorname{diag}}\left(\frac{a_{2}-a_{1}}{kn_{1}},\ldots,\frac{a_{2}-a_{1}}{kn_{1}},\frac{a_{3}-a_{2}}{kn_{2}},\ldots,\frac{a_{3}-a_{2}}{kn_{2}},\ldots,\frac{a_{k}-a_{k-1}}{kn_{k-1}},\ldots,\frac{a_{k}-a_{k-1}}{kn_{k-1}},\right.\\ \left.\frac{a_{1}-a_{k}}{kn_{k}},\ldots,\frac{a_{1}-a_{k}}{kn_{k}}\right).$ Then $W$ has trace zero and $\mathcal{D}A+W={\operatorname{diag}}\left(\frac{(n_{1}-1)a_{1}+a_{2}}{n_{1}},\ldots,\right.\\ \left.\frac{(n_{1}-1)a_{1}+a_{2}}{n_{1}},\right.$ $\left.\frac{(n_{2}-1)a_{2}+a_{3}}{n_{2}},\ldots,\frac{(n_{2}-1)a_{2}+a_{3}}{n_{2}},\ldots,\frac{(n_{k}-1)a_{k-1}+a_{k}}{n_{k}},\ldots,\frac{(n_{k}-1)a_{k-1}+a_{k}}{n_{k}}\right).$ It is easy to check that $\|\mathcal{D}A+W\|<1$. Hence for this particular $\mathscr{W}$ we have that $\{A:\|A+W\|\geq\|A\|\text{ for all }W\in\mathscr{W}\}=\mathscr{W}^{\perp}=\mathbb{C}I$. 4. 4.

In Theorem 1, $m(A)$ is the best possible upper bound on rank $P$. Consider $\mathscr{W}=\{X:{\operatorname{tr\ }}X=0\}$. From Remark 2, we get that if a matrix $A$ is orthogonal to $\mathscr{W}$ then it has to be of the form $A=\lambda I$, for some $\lambda\in\mathbb{C}$. When $A\neq 0$ then $m(A)=n$. Let $P$ be any density matrix satisfying $AP\in\mathscr{W}^{\perp}$. Then $AP=\mu I$, for some $\mu\in\mathbb{C},\mu\neq 0$. If $P$ also satisfies $A^{*}AP=\|A\|^{2}P$, then we get $P=\frac{\mu}{\lambda}I$. Hence rank $P=n=m(A)$. 5. 5.

For $n=2$ and $\mathscr{B}$ any subalgebra of $\mathbb{M}_{2}(\mathbb{C})$, we can restrict maximum on the right hand side of (5) over rank one density matrices. By the same argument as in the proof of Theorem 2 it is sufficient to prove this for $\mathbb{D}_{2}(\mathbb{C})$, the subalgebra of diagonal matrices with complex entries. We show

[TABLE]

where $\Delta$ is the projection onto $\mathbb{D}_{2}(\mathbb{C})$. From Theorem 2 we have

[TABLE]

Note that

[TABLE]

Let A=\left[\begin{array}[]{ccc}a&b\\ c&d\end{array}\right] and without loss of generality assume that $|b|\geq|c|$. Then $\|\mathcal{O}A\|=|b|$. For x=\left[\begin{array}[]{ccc}0\\ 1\end{array}\right]

[TABLE]

Combining this with (33), we obtain

[TABLE] 6. 6.

For $n=2$ and $\mathscr{B}$ any subalgebra of $\mathbb{M}_{2}(\mathbb{C})$, we note that

[TABLE]

Again it is enough to show that

[TABLE]

If $A$ is an off-diagonal $2\times 2$ matrix, that is, A=\left[\begin{array}[]{ccc}0&b\\ c&0\end{array}\right] then by Theorem 2.1 in [5] we obtain $\|A+W\|\geq\|A\|\text{ for all }W\in\mathbb{D}_{2}(\mathbb{C})$. Conversely let $A\in\mathbb{M}_{n}(\mathbb{C})$ be such that $\|A+W\|\geq\|A\|\text{ for all }W\in\mathbb{D}_{2}(\mathbb{C})$. Then by taking $W=-\mathcal{D}A$, we have $A+W=\mathcal{O}(A)$. Again by using Theorem 2.1 in [5] we obtain that $\|\mathcal{O}(A)\|=\|A\|$. So $A$ is of the form \left[\begin{array}[]{ccc}a&b\\ c&d\end{array}\right], where $\|A\|=\max\{|b|,|c|\}.$ Since norm of each row and each colum is less than or equal to $\|A\|$, we get that $a=d=0$. Hence $A\in\mathbb{D}_{2}(\mathbb{C})^{\perp}$.

Acknowledgement. I would like to thank Professor Rajendra Bhatia for several useful discussions and Professor Ajit Iqbal Singh for helpful comments in this paper.