On the Distribution of GSVD

TL;DR

This paper presents new theoretical results concerning the distribution of the generalized singular value decomposition (GSVD), enhancing understanding of its statistical properties.

Contribution

It introduces novel findings on the distribution of GSVD, contributing to the theoretical foundation of this matrix decomposition method.

Findings

New distributional properties of GSVD derived

Enhanced understanding of GSVD behavior

Theoretical insights into GSVD applications

Abstract

In this paper, some new results on the distribution of the generalized singular value decomposition (GSVD) are presented.

Equations

$$\mathbf{UAQ}=\left({\mathbf{\Sigma }}_{\mathbf{A}},\mathbf{O}\right)\text{and}\mathbf{VCQ}=\left({\mathbf{\Sigma }}_{\mathbf{C}},\mathbf{O}\right),$$

$$\displaystyle\mathbf{\Sigma}_{\mathbf{A}}=\left(\begin{array}[]{ccc}\mathbf{I}_{r}&&\\ &\mathbf{S}_{\mathbf{A}}&\\ &&\mathbf{O}_{\mathbf{A}}\end{array}\right)\quad\text{and}\quad\mathbf{\Sigma}_{\mathbf{C}}=\left(\begin{array}[]{ccc}\mathbf{O}_{\mathbf{C}}&&\\ &\mathbf{S}_{\mathbf{C}}&\\ &&\mathbf{I}_{k-r-s}\end{array}\right),$$

$$\mathbf{L}={\mathbf{X}}^H{\left(\mathbf{Y}{\mathbf{Y}}^H\right)}^{-1}\mathbf{X}.$$

$$\displaystyle(m^{\prime},p,n^{\prime})=\left\{\begin{array}[]{ll}(n,m,q)&q\geq n\\ (q,s,n)&q<n<(q+m)\end{array}\right..$$

$$f_{m^{\prime },p,n^{\prime }}(w_1,\cdots ,w_l)={\mathbf{M}}_{m^{\prime },p,n^{\prime }}\frac{{\prod }_{i=1}^l{w}_i^{t_1}}{{\prod }_{i=1}^l(1+w_i{)}^{t_2}}\prod _{i<j}^l(w_i-w_j{)}^2,$$

$$\displaystyle\mathbf{M}_{m^{\prime},p,n^{\prime}}=\left\{\begin{array}[]{ll}\frac{\pi^{m^{\prime}(m^{\prime}-1)}\widetilde{\Gamma}_{m^{\prime}}(p+n^{\prime})}{m^{\prime}!\widetilde{\Gamma}_{m^{\prime}}(p)\widetilde{\Gamma}_{m^{\prime}}(n^{\prime})\widetilde{\Gamma}_{m^{\prime}}(m^{\prime})}&p\geq m^{\prime}\\ \frac{\pi^{p(p-1)}\widetilde{\Gamma}_{p}(p+n^{\prime})}{p!\widetilde{\Gamma}_{p}(m^{\prime})\widetilde{\Gamma}_{p}(p+n^{\prime}-m^{\prime})\widetilde{\Gamma}_{p}(p)}&p<m^{\prime}\end{array}\right.,$$

$$g_{m^{\prime },p,n^{\prime }}(w_l)={\mathbf{M}}_{m^{\prime },p,n^{\prime }}{g}_{l,t_1,t_2}^{\prime }(w_l)\text{or}\frac{1}{w_l^2}{\mathbf{M}}_{m^{\prime },p,n^{\prime }}{g}_{l,t_1^{\prime },t_2}^{\prime }(1/w_l),$$

$$g_{l,t_1,t_2}^{\prime }(w_l)$$

$$\displaystyle\mathbf{Q}=\mathbf{Q}^{\prime}\left(\begin{array}[]{cc}\left(\mathbf{W}^{H}\mathbf{R}\right)^{-1}&\\ &\mathbf{O}_{n-k}\end{array}\right),$$

$$\text{trace}\{\mathbf{Q}{\mathbf{Q}}^H\}$$

$$=$$

$$\mathcal{E}\{\text{trace}\{\mathbf{Q}{\mathbf{Q}}^H\}\}=\frac{\min \{m+q,n\}}{|m+q-n|}.$$

$$\mathbf{UAQ}={\mathbf{\Sigma }}_{\mathbf{A}}\text{and}\mathbf{VCQ}={\mathbf{\Sigma }}_{\mathbf{C}}.$$

$${\mathbf{A}}^H\mathbf{A}={\mathbf{Q}}^{-H}{\mathbf{\Sigma }}_{\mathbf{A}}^H{\mathbf{\Sigma }}_{\mathbf{A}}{\mathbf{Q}}^{-1}\text{and}{\mathbf{C}}^H\mathbf{C}={\mathbf{Q}}^{-H}{\mathbf{\Sigma }}_{\mathbf{C}}^H{\mathbf{\Sigma }}_{\mathbf{C}}{\mathbf{Q}}^{-1}.$$

$$\displaystyle\mathbf{\Sigma}_{\mathbf{A}}^{H}\mathbf{\Sigma}_{\mathbf{A}}=\left(\begin{array}[]{cc}\mathbf{S}_{\mathbf{A}}^{H}\mathbf{S}_{\mathbf{A}}&\\ &\mathbf{O}_{n-s}\end{array}\right)\quad\text{and}\quad\mathbf{\Sigma}_{\mathbf{C}}^{H}\mathbf{\Sigma}_{\mathbf{C}}=\left(\begin{array}[]{cc}\mathbf{S}_{\mathbf{C}}^{H}\mathbf{S}_{\mathbf{C}}&\\ &\mathbf{I}_{n-s}\end{array}\right).$$

$$\displaystyle\left(\mathbf{C}^{H}\mathbf{C}\right)^{-1}\mathbf{A}^{H}\mathbf{A}=\mathbf{Q}\left(\begin{array}[]{cc}\mathbf{S}_{\mathbf{A}}^{H}\mathbf{S}_{\mathbf{A}}\left[\mathbf{S}_{\mathbf{C}}^{H}\mathbf{S}_{\mathbf{C}}\right]^{-1}&\\ &\mathbf{O}_{n-s}\end{array}\right)\mathbf{Q}^{-1}.$$

$$\mathbf{L}$$

$$\displaystyle\mathbf{P}=\left(\begin{array}[]{cc}\mathbf{P}_{11}&\mathbf{P}_{12}\\ \mathbf{P}_{21}&\mathbf{P}_{22}\end{array}\right),$$

$$\mathbf{P}{\mathbf{P}}^H={\mathbf{P}}^H\mathbf{P}={\mathbf{I}}_{m+q}.$$

$${\mathbf{P}}_{11}{\mathbf{P}}_{11}^H+{\mathbf{P}}_{12}{\mathbf{P}}_{12}^H={\mathbf{I}}_m\text{and}{\mathbf{P}}_{22}^H{\mathbf{P}}_{22}+{\mathbf{P}}_{12}^H{\mathbf{P}}_{12}={\mathbf{I}}_{m+q-n}.$$

$$\displaystyle\mathbf{P}^{\prime}=\left(\begin{array}[]{cc}\mathbf{P}_{22}^{H}&\mathbf{P}_{12}^{H}\\ \mathbf{P}_{21}^{H}&\mathbf{P}_{11}^{H}\end{array}\right).$$

$$g_{m^{\prime },p,n^{\prime }}(w_l)$$

$$g_{l,t_1,t_2}^{\prime }(w_l)$$

$$\prod _{i<j}^l(w_i-w_j{)}^2$$

$$=$$

$$\prod _{i<j}^l(w_i-w_j{)}^2=|\mathbf{W}{|}^2=\sum _{{\sigma }_1,{\sigma }_2\in S_l}\mathbf{sign}({\sigma }_1)\mathbf{sign}({\sigma }_2)\prod _{i=1}^l{w}_i^{2l-{\sigma }_1(i)-{\sigma }_2(i)}.$$

$$g_{l,t_1,t_2}^{\prime }(w_l)$$

$$g_{l,t_1,t_2}^{\prime }(1/w_l)$$

Full text

I Some new results on GSVD

In this section, the GSVD of two Gaussian matrices is defined first. Then, the distribution of the squared generalized singular values are presented.

I-A Definition of GSVD

Given two matrices $\mathbf{A}\in\mathbb{C}^{m\times n}$ and $\mathbf{C}\in\mathbb{C}^{q\times n}$, whose entries are i.i.d. complex Gaussian random variables with zero mean and unit variance. Let us define $k=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m+q,n\}$, $r=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})-\text{rank}(\mathbf{C})=\min\{m+q,n\}-\min\{q,n\}$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}$. Then, the GSVD of $\mathbf{A}$ and $\mathbf{C}$ can be expressed as follows [1]:

[TABLE]

where $\mathbf{\Sigma}_{\mathbf{A}}\in\mathbb{C}^{m\times k}$ and $\mathbf{\Sigma}_{\mathbf{C}}\in\mathbb{C}^{q\times k}$ are two nonnegative diagonal matrices, $\mathbf{U}\in\mathbb{C}^{m\times m}$ and $\mathbf{V}\in\mathbb{C}^{q\times q}$ are two unitary matrices, and $\mathbf{Q}\in\mathbb{C}^{n\times n}$ can be expressed as in (21).

Moreover, $\mathbf{\Sigma}_{\mathbf{A}}$ and $\mathbf{\Sigma}_{\mathbf{C}}$ have the following form:

[TABLE]

where $\mathbf{S}_{\mathbf{A}}=\textrm{diag}(\alpha_{1},\cdots,\alpha_{s})$ and $\mathbf{S}_{\mathbf{C}}=\textrm{diag}(\beta_{1},\cdots,\beta_{s})$ are two $s\times s$ nonnegative diagonal matrices, satisfying $\mathbf{S}_{\mathbf{A}}^{2}+\mathbf{S}_{\mathbf{C}}^{2}=\mathbf{I}_{s}$. Then, the squared generalized singular values can be defined as $w_{i}=\alpha_{i}^{2}/\beta_{i}^{2}$, $i\in\{1,\cdots,s\}$.

I-B Distribution of the squared generalized singular values

To characterize the distribution of the squared generalized singular value $w_{i}$, a relationship between $w_{i}$ and the eigenvalue of a common matrix model is established first as in the following theorem.

Theorem 1

Suppose that $\mathbf{A}\in\mathbb{C}^{m\times n}$ and $\mathbf{C}\in\mathbb{C}^{q\times n}$ are two Gaussian matrices whose elements are i.i.d. complex Gaussian random variables with zero mean and unit variance and their GSVD is defined as in (1). Without loss of generality, it is assumed that $q\geq m$. Then, the distribution of their squared generalized singular values, $w_{i},i\in\{1,\cdots,s\}$, is identical to that of the nonzero eigenvalues of $\mathbf{L}$, where

[TABLE]

$\mathbf{X}\in\mathbb{C}^{m^{\prime}\times p}$* and $\mathbf{Y}\in\mathbb{C}^{m^{\prime}\times n^{\prime}}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. Moreover, $m^{\prime}$, $p$ and $n^{\prime}$ can be expressed as follows:*

[TABLE]

When $(q+m)\leq n$, $s=0$ and $\mathbf{\Sigma}_{\mathbf{A}}$ and $\mathbf{\Sigma}_{\mathbf{C}}$ are deterministic.

Proof:

See Appendix A. ∎

The distribution of the nonzero eigenvalues of $\mathbf{L}$ can be characterized as in the following corollary.

Corollary 1

Suppose that $\mathbf{L}=\mathbf{X}^{H}\left(\mathbf{Y}\mathbf{Y}^{H}\right)^{-1}\mathbf{X}$, and $\mathbf{X}\in\mathbb{C}^{m^{\prime}\times p}$ and $\mathbf{Y}\in\mathbb{C}^{m^{\prime}\times n^{\prime}}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. And, it is assumed that $m^{\prime}\leq n^{\prime}$. Then, the joint probability density function (p.d.f.) of the nonzero eigenvalues of $\mathbf{L}$ can be characterized as follows:

[TABLE]

where $l=\min\{p,m^{\prime}\}$, $t_{1}=|m^{\prime}-p|$, $t_{2}=p+n^{\prime}$, and $\mathbf{M}_{m^{\prime},p,n^{\prime}}$ can be expressed as follows:

[TABLE]

where $m^{\prime}!=1\times 2\times\cdots m^{\prime}$ and $\widetilde{\Gamma}_{m^{\prime}}(p)$ is the complex multivariate gamma function[2].

Proof:

Following steps similar to those in [3, Appendix A], the distribution of the nonzero eigenvalues of $\mathbf{L}$ can be obtained. ∎

Moreover, the marginal p.d.f. of $w_{i}$, $i\in\{1,\cdots,l\}$, can be characterized as in the following lemma.

Lemma 1

The marginal p.d.f. derived from (13) can be expressed as follows:

[TABLE]

where $t^{\prime}_{1}=n^{\prime}-m^{\prime}$ and $g^{\prime}_{l,t_{1},t_{2}}(w_{l})$ can be expressed as follows:

[TABLE]

where $\sigma_{1}=\left(\sigma_{1}(1),\sigma_{1}(2)\cdots\sigma_{1}(l)\right)$, $\sigma_{2}=\left(\sigma_{2}(1),\sigma_{2}(2)\cdots\sigma_{2}(l)\right)\in S_{l}$ are permutations of length $l$, $\mathbf{sign}(*)$, $(*)\in\{\sigma_{1},\sigma_{2}\}$, is $1$ if the permutation is even and $-1$ if it is odd, and $B(t_{1}+2l-\sigma_{1}(i)-\sigma_{2}(i)+1,t_{2}-t_{1}-2l-1+\sigma_{1}(i)+\sigma_{2}(i))$ is the Beta function [4].

Proof:

See Appendix B. ∎

I-C Some properties about $\mathbf{Q}$

As shown in [1], the GSVD decomposition matrix $\mathbf{Q}$ is often used to construct the precoding matrix at the transmitting end. In this section, some properties about $\mathbf{Q}$ are discussed.

First, define $\mathbf{B}=(\mathbf{A}^{H},\mathbf{C}^{H})^{H}$. As shown in [5, eq.(2.2)], $\mathbf{Q}$ can be expressed as

[TABLE]

where $\mathbf{Q}^{\prime}\in\mathbb{C}^{n\times n}$ and $\mathbf{W}\in\mathbb{C}^{k\times k}$ are two unitary matrices, $\mathbf{R}\in\mathbb{C}^{k\times k}$ is a nonnegative diagonal matrix and has the same singular values as the nonzero singular values of $\mathbf{B}$. Thus, the power of $\mathbf{Q}$ can be expressed as

[TABLE]

where $\lambda_{B,i}$, $i\in\{1,\cdots,k\}$, are the nonzero eigenvalues of $\mathbf{B}\mathbf{B}^{H}$.

Note that $\mathbf{A}$ and $\mathbf{C}$ are two independent Gaussian matrices. Then, it is easy to know that $\mathbf{B}\mathbf{B}^{H}$ is a Wishart matrix. Finally, directly from [6, Lemma 2.10], the following corollary can be derived.

Corollary 2

Suppose that $\mathbf{A}\in\mathbb{C}^{m\times n}$ and $\mathbf{C}\in\mathbb{C}^{q\times n}$ are two Gaussian matrices whose elements are i.i.d. complex Gaussian random variables with zero mean and unit variance and their GSVD is defined as in (1). The average power of the GSVD decomposition matrix $\mathbf{Q}$ can be expressed as follows:

[TABLE]

Appendix A: Proof of Theorem 1

1. The case when $q\geq n$

When $q\geq n$, $k=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m+q,n\}=n$, $r=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})-\text{rank}(\mathbf{C})=\min\{m+q,n\}-\min\{q,n\}=0$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}=\min\{m,n\}$. Thus, from (1), the GSVD of $\mathbf{A}$ and $\mathbf{C}$ can be further expressed as follows:

[TABLE]

Moreover, when $k=n$, as shown in (21), $\mathbf{Q}$ is nonsingular and it can be shown that

[TABLE]

Furthermore, from (8), it can be shown that

[TABLE]

Thus, $\left(\mathbf{C}^{H}\mathbf{C}\right)^{-1}\mathbf{A}^{H}\mathbf{A}$ can be expressed as

[TABLE]

Recall that $\mathbf{S}_{\mathbf{A}}=\textrm{diag}(\alpha_{1},\cdots,\alpha_{s})$ and $\mathbf{S}_{\mathbf{C}}=\textrm{diag}(\beta_{1},\cdots,\beta_{s})$. Then, it can be shown that $\mathbf{S}_{\mathbf{A}}^{H}\mathbf{S}_{\mathbf{A}}\left[\mathbf{S}_{\mathbf{C}}^{H}\mathbf{S}_{\mathbf{C}}\right]^{-1}=\textrm{diag}(\alpha_{1}^{2}/\beta_{1}^{2},\cdots,\alpha_{s}^{2}/\beta_{s}^{2})=\textrm{diag}(w_{1},\cdots,w_{s})$. Finally, it is easy to see that the distribution of $w_{i},i\in\{1,\cdots,s\}$, is identical to that of the nonzero eigenvalues of $\mathbf{L}$, where

[TABLE]

$\mathbf{X}\in\mathbb{C}^{m^{\prime}\times p}=\mathbf{A}^{H}$ and $\mathbf{Y}\in\mathbb{C}^{m^{\prime}\times n^{\prime}}=\mathbf{C}^{H}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance. Moreover, $(m^{\prime},p,n^{\prime})=(n,m,q)$.

2. The case when $q<n<(q+m)$

When $q<n<(q+m)$, $k=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m+q,n\}=n$, $r=\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})-\text{rank}(\mathbf{C})=\min\{m+q,n\}-\min\{q,n\}=n-q$, and $s=\text{rank}(\mathbf{A})+\text{rank}(\mathbf{C})-\text{rank}((\mathbf{A}^{H},\mathbf{C}^{H})^{H})=\min\{m,n\}+\min\{q,n\}-\min\{m+q,n\}=m+q-n$. As shown in (8), $w_{1},\cdots,w_{s}$ are the squared generalized singular values of $\mathbf{A}$ and $\mathbf{C}$, and $\alpha_{i}^{2}=\frac{w_{i}}{1+w_{i}},i\in\{1,\cdots,s\}$.

On the other hand, define $\mathbf{B}=(\mathbf{A}^{H},\mathbf{C}^{H})^{H}$ and the SVD of $\mathbf{B}$ as $\mathbf{B}=\mathbf{P}\mathbf{\Sigma}_{\mathbf{B}}\mathbf{R}$. Note that $\mathbf{P}\in\mathbb{C}^{(m+q)\times(m+q)}$ is a Haar matrix. Divide $\mathbf{P}$ into the following four blocks:

[TABLE]

where $\mathbf{P}_{11}\in\mathbb{C}^{m\times n}$ and $\mathbf{P}_{22}\in\mathbb{C}^{q\times(m+q-n)}$. From [5, eq.(2.7)], it is easy to see that $\alpha_{1}^{2},\cdots,\alpha_{s}^{2}$ equal the non-one eigenvalues of $\mathbf{P}_{11}\mathbf{P}_{11}^{H}$. Moreover, from the fact that $\mathbf{P}$ is a Haar matrix, it can be shown that

[TABLE]

Thus, the following equations can be derived:

[TABLE]

Note that $\mathbf{P}_{12}\mathbf{P}_{12}^{H}$ and $\mathbf{P}_{12}^{H}\mathbf{P}_{12}$ have the same non-zero eigenvalues. Thus, $\mathbf{P}_{11}\mathbf{P}_{11}^{H}$ and $\mathbf{P}_{22}^{H}\mathbf{P}_{22}$ have the same non-one eigenvalues. Therefore, $\alpha_{1}^{2},\cdots,\alpha_{s}^{2}$ equal the non-one eigenvalues of $\mathbf{P}_{22}^{H}\mathbf{P}_{22}$. Define $\mathbf{P}^{\prime}$ as

[TABLE]

It is easy to see that $\mathbf{P}^{\prime}\mathbf{P}^{\prime H}=\mathbf{P}^{\prime H}\mathbf{P}^{\prime}=\mathbf{I}_{m+q}$ and $\mathbf{P}^{\prime}$ is also a Haar matrix.

Then, from the above discussions, it can be concluded that the distribution of $\alpha_{1},\cdots,\alpha_{s}$ is identical to the distribution of the non-one singular valus of the $m\times n$ or $(m+q-n)\times q$ truncated sub-matrix of a $(m+q)\times(m+q)$ Haar matrix. Thus, define $\mathbf{A}^{\prime}\in\mathbb{C}^{(m+q-n)\times q}$ and $\mathbf{C}^{\prime}\in\mathbb{C}^{n\times q}$, whose entries are i.i.d. complex Gaussian random variables with zero mean and unit variance. The distribution of the squared squared generalized singular values of $\mathbf{A}$ and $\mathbf{C}$, is identical to that of the squared squared generalized singular values of $\mathbf{A}^{\prime}$ and $\mathbf{C}^{\prime}$. Since $n>q$, from Appendix A-1, it can be known that the distribution of the squared squared generalized singular values of $\mathbf{A}^{\prime}$ and $\mathbf{C}^{\prime}$, is identical to that of the eigenvalues of $\mathbf{L}=\mathbf{X}^{H}\left(\mathbf{Y}\mathbf{Y}^{H}\right)^{-1}\mathbf{X}$, where $\mathbf{X}\in\mathbb{C}^{q\times(m+q-n)}$ and $\mathbf{Y}\in\mathbb{C}^{q\times n}$ are two independent Gaussian matrices whose elements are are i.i.d. complex Gaussian random variables with zero mean and unit variance.

This completes the proof of the theorem. $\blacksquare$

Appendix B: Proof of Lemma 1

First, the marginal p.d.f. derived from (13) can be expressed as follows:

[TABLE]

where $f^{\prime}_{l,t_{1},t_{2}}(w_{1},\cdots,w_{l})dw_{1}\cdots dw_{l-1}=\frac{\prod_{i=1}^{l}w_{i}^{t_{1}}}{\prod_{i=1}^{l}(1+w_{i})^{t_{2}}}\prod_{i<j}^{l}(w_{i}-w_{j})^{2}$, and

[TABLE]

Note that $\prod_{i<j}^{l}(w_{i}-w_{j})^{2}$ can be expressed as

[TABLE]

Thus, $\prod_{i<j}^{l}(w_{i}-w_{j})^{2}$ can be further expressed as

[TABLE]

Therefore, $g^{\prime}_{l,t_{1},t_{2}}(w_{l})$ can be expressed as

[TABLE]

Moreover, from Eq.(3.194.3) [4], it can be shown that $\int_{0}^{\infty}\frac{w_{i}^{t_{1}+2l-\sigma_{1}(i)-\sigma_{2}(i)}}{(1+w_{i})^{t_{2}}}dw_{i}=B(t_{1}+2l-\sigma_{1}(i)-\sigma_{2}(i)+1,t_{2}-t_{1}-2l-1+\sigma_{1}(i)+\sigma_{2}(i)).$ Then, $g^{\prime}_{l,t_{1},t_{2}}(w_{l})$ can be expressed as

[TABLE]

On the other hand, as shown in (Appendix B: Proof of Lemma 1), $g^{\prime}_{l,t_{1},t_{2}}(1/w_{l})$ can be be expressed as

[TABLE]

Moreover, define $w^{\prime}_{i}=\frac{1}{w_{i}}$, $i\in\{1,\cdots,l-1\}$. Then, $g^{\prime}_{l,t_{1},t_{2}}(1/w_{l})$ can be be further expressed as

[TABLE]

where $t_{1}^{\prime}=t_{2}-t_{1}-2l=n^{\prime}-m^{\prime}$. Thus, it can be known that $g_{m^{\prime},p,n^{\prime}}(w_{l})=\mathbf{M}_{m^{\prime},p,n^{\prime}}g^{\prime}_{l,t_{1},t_{2}}(w_{l})=\mathbf{M}_{m^{\prime},p,n^{\prime}}(1/w_{l})^{2}g^{\prime}_{l,t_{1}^{\prime},t_{2}}(1/w_{l})$.

This completes the proof of the lemma. $\blacksquare$