Questions tagged [matrix-analysis]

For question about matrices and their algebraic properties. Together with [tag:linear-algebra] if necessary.

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The necessary and sufficient constraint of $A$ such that $\text{tr}(A^n)\geq0$?

I'm looking for a sufficient and necessary constraint for the complex square matrix $A$ such that $\text{tr}(A^n)\geq0$ for $n=1,2,3...$ Up to now, it seems that the eigenvalues of $A$ must come in ...
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1 answer
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The minimal polynomial of a matrix

Let $A=(a_{kl})$ be the matrix in $M_n(\mathbb{C})$ given by $a_{kk}=0$ and $a_{kl}=\frac12$ if $k\neq l$. Let $D$ be the diagonal matrix in $M_n(\mathbb{C})$ with $D=\operatorname{diag}(n,\frac{2n+...
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1 vote
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How do I calculate the derivatives of the following equation? [closed]

$\frac{ \partial }{ \partial \mathbf{F} } \mathrm{tr} \{ \mathbf{T}^{H} \mathbf{F} ( \mathbf{F}^{H} \mathbf{F} )^{-1} \mathbf{F}^{H} \mathbf{T}\}$, where $\mathbf{F} \in \mathbb{C}^{N \times M}, \...
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2 answers
172 views

Projection onto the Stiefel manifold and the orthogonal Procrustes problem

Let $m$ and $n$ be positive integers such that $m \ge n$, the case with $m >n$ being particularly interesting to us. The Stiefel manifold is \begin{equation} \mathbb{S}^{m, n} = \{X \in \mathbb{R}^{...
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Bhatia, Ex 7.1.12: Principal angles between subspaces

I am unable to solve this problem in Matrix Analysis by Rajendra Bhatia: I can't see how to prove that $\theta_i$ are decreasing. I also have no intuition for why the angles are defined this way. ...
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  • 467
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1 answer
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Property regarding two positive definite matrices

Suppose $A$ and $B$ are two $n \times n$ symmetric positive definite real matrices and let $M$ be an $n \times m$ real matrix of full rank. Suppose that $A - B$ is positive definite. Then show that ...
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If $T=\begin{bmatrix}x&Y^*\\Y &Z\end{bmatrix}$, $Z\ge 0$ $n\times n$ matrix, $x>0$ and $Y$ is $n\times 1$ vector, $T\ge 0$ iff $Y\in\text{ran }(Z)$

This is what I have to prove: If $T=\begin{bmatrix} x & Y^* \\ Y & Z\end{bmatrix}$ where $Z$ is positive semidefinite $n\times n$ matrix, $x>0$ and $Y$ is $n\times 1$ column vector then $T$...
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Convexity of a special kind of set

Let $\mathcal{H}$ be a separable Hilbert space and $T\in B(\mathcal{H})$. Is the following set convex \begin{align} \{\langle Te_1,e_2\rangle:\{e_1,e_2\} \text{ are orthonormal set in } \mathcal{H}\}? ...
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  • 356
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Convertor matrices

Let us consider $\mathbb{C}^n$, the set of all $n$-tuples of complex numbers. We know that $(\mathbb{C}^n,+,\cdot)$ is a unital commutative complex algebra. Suppose that $\diamond$ is an ...
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Does $\|A^k\| \leq a k^{n-1} \rho^k(A)$ hold?

Given a non-singular matrix $A \in \mathbb{R}^{n\times n}$, let $\rho(A)$ be its spectral radius. Then, $\forall k \in \mathbb{N_0}$, can we find a constant $a > 0$ such that the following ...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this link for detail), i.e., say $a$ is a complex ...
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1 answer
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Proof that sum of k-eigenvalues is convex

I saw a post https://mathoverflow.net/questions/98367/a-sum-of-eigenvalues that said It is well-known that $\sum^{r}_{i = 1} \lambda_{i}(X)$ is convex. and I saw an explanation in Boyd and ...
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1 answer
38 views

Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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Proof the limit of matrices means about $e^H$ (where matrix $H$ is self-adjoint)

This problem is from Chap.6 of Introduction to Matrix Analysis and Applications of Petz. Prove for self-adjoint matrices $H$, $K$ that $$ \lim _{r \rightarrow 0} \left(e^{r H} \#_{\alpha}e^{r K}\...
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1 vote
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Uniform convergence under matrix inverse

Let $A_n\in\mathbb{R}^{d\times d}$ be a sequence of positive definite matrices such that $\sigma_{\text{min}}(A_n) = 1/n$ and $\sigma_{\text{max}}(A_n) = 1$ for all $n\in\mathbb{N}$. Let $X\in\mathbb{...
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  • 33
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1 answer
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Weighted least squares formula

I was studying the weighted least squares algorithm and came across this formula for calculating the weighted result in terms of the original. Here $x_{LS}$ is the solution for $D=I$ I can't figure ...
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Inequalities that relates the norms of $AB^{-1}$ and $A-B$?

Let $A,B$ be two $d\times d$ invertible real matrices. Are there any famous matrix inequalities relating the norms (any norm since all norms over finite dimeonsion spaces are equivalent) of $AB^{-1}$ ...
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Matrix of logarithmic mean

I want to prove the matrix logarithmic mean in differential form. $ L(A, B)^{-1}=\int_{0}^{\infty} \frac{(t A+B)^{-1}}{t+1} d t $ for positive definite matrices $A, B$. I know $L(A, B)^{-1}=\frac{\log ...
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3 votes
2 answers
116 views

Is there a way to calculate the eigenvalues of $xx^T+yy^T$?

As the title shows, is there a way to calculate the eigenvalues of $A\equiv \vec x\vec x^T+\vec y \vec y^T$, where $\vec x$ and $\vec y$ are two linearly independent vectors in $\mathbb{R}^n$(don't ...
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Spectral Radius of a non negative matrix

Let $A \in \mathbb{R}^{n\times n}$ be a non-negative square matrix and $x \in \mathbb{R}^{n}$ be a positive eigenvector. Prove that \begin{gather*} \rho(A)=\max\limits_{x>0}\; \min\limits_{i=1,...
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Bound a matrix given its bounded product

We are given a full rank square matrix $A \in\Bbb R^{d\times d}$ and a vector $x \in\Bbb R^{d \times 1}$, we know that \begin{align*} \|Ax\| \le C \end{align*} Is there a way to construct an upper ...
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1 vote
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Norm estimate of a special kind of 2-by-2 matrix

Let $S\subseteq\mathbb{T}=\{z\in\mathbb{C}:\vert z\vert=1\}$ with $\text{conv}(S)$ contains $\left\{z\in\mathbb{C}:\vert z\vert\leq\frac{1}{\sqrt{2}}\right\}$. Define $$\mathcal{A}_S:=\{B\in M_2: \Re(\...
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-3 votes
1 answer
72 views

If $AB^∗$ and $B^∗A$ are both normal, show that $B A^∗A = A A^∗B$ [closed]

This might be an easy question, but I haven't been able to prove it. Prove that if $A B^*$ and $B^* A$ are normal matrices, then $B A^* A = A A^* B$ Any help is appreciated. Here, $A^*$ is the ...
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1 answer
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What's the singular value of a symmetric matrix plus identy matrix? $A+\lambda I$

Suppose, we know the singular values of a symmetric matrix $A$ as $\{\sigma_1,\cdots,\sigma_n\}$. What is the singular values of the matrix $A+\lambda I$?
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1 vote
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Prove that if a function is 2-monotone on $(a, c]$ and $[c, b)$, then it is 2-monotone on $(a, b)$

I am studying matrix monotone and I meet some difficulties when trying to prove the following statement: Let $f$ be a differentiable function on the interval $(a,b)$ such that $f$ is 2-monotone on $(a,...
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1 vote
0 answers
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Algebraic relation between symmetric matrix and its principal submatrix?

Let $A$ be a $n\times n$ real symmetric matrix and $B$ be $m\times m$ real symmetric matrix where $n>m$. $B$ be a principal submatrix of $A$ (i.e) (obtained by deleting both $i$-th row and $i$-th ...
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0 answers
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how to prove that eigenvalues of a matrix is in unit disk

Define block matrix with real entries $$A=\begin{bmatrix}I-\alpha H-\alpha\beta L &-\beta I \\ \alpha L & I \end{bmatrix}$$ where $H,L$ are real symmetric positive semi-definite (may be ...
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1 answer
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How can I map the matrix monotone to completely monotone function?

I currently study matrix monotone. I met some trouble with the proof below. Prove if $f: \mathbb{R}^{+} \rightarrow \mathbb{R}$ is a matrix monotone function, then $-f$ is a completely monotone ...
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1 answer
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Spectral norm of matrix restricted to subspace $1^\perp$

On Page 6 (of 23) of the following document: Boyd et. al - Fastest Mixing Markov Chain on a Graph The spectral norm of a matrix P, restricted to subspace $\mathbf{1}^\perp=\{u \in R^{n} \mid \mathbf{1}...
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0 answers
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Lower bound of the smallest singular value of product of two matrices

Suppose that $X, Y \in \mathbb{R}^{m \times n}$ both have full row rank ($m < n$), is it possible to obtain the lower bound of $\sigma_{\min} (XY^{T})$? The smallest non-zero singular value of AB ...
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3 votes
2 answers
134 views

Prove the following matrix commute with every matrix

This question is from A Second Course in Linear Algebra. Some appropriate terminologies are defined as follows: Definition 1. Let $V$ be a finite-dimensional complex inner product space over and $\...
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1 answer
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What is the relationship between $A^{-1}$, ${\bigwedge}^n A$, and ${\bigwedge}^{n-1} A$?

I am stumped on this one problem mentioned in Bhatia's Matrix Analysis book: For an invertible operator, obtain a relationship between $A^{-1}$, ${\bigwedge}^n A$, and ${\bigwedge}^{n-1} A$. The ...
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1 answer
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Divided differences problem in order n.

This problem is in Chap.3 of Introduction to Matrix Analysis and Applications of Petz. If $f (x) = x^k$ with $k \in \mathbb{N}$, verify that $$ f^{[n]}\left[x_{1}, x_{2}, \ldots, x_{n+1}\right] = \...
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1 vote
1 answer
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A and B have identical singular-values and identical eigenvalues, are they unitary similar?

I've come across this question while taking a matrix analysis course. Given: $A,B \in\mathbb{C}^{3 \times 3}$ with $\lambda_1(A) = \lambda_1(B) ,\,\lambda_2(A) = \lambda_2(B),\,\lambda_3(A) = \...
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2 votes
2 answers
172 views

When is $A^TBA$ invertible, where $B$ is an invertible, symmetric matrix? [closed]

Let $B$ be an $n×n$ invertible matrix such that $B^T=B$, and let $A$ be a $n×2$ matrix with linearly independent columns. When is the product $A^TBA$ invertible?
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How to prove spectral radius of $A-A^{\infty}$ is less than 1? $A$ is semi convergent

Suppose that $A$ is a semi-convergent matrix. Denote the limiting matrix as $A^{\infty}$. Can someone tell me, why the spectral radius of ($A-A^{\infty}$) is less than 1? I guess it is because $A-A^{\...
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0 votes
2 answers
56 views

What is the solution to this matrix optimization problem $A^* = \text{argmin}_{A} \sum_{i=1}^{r-1}|Ax_i-x_{i+1}|^2$?

The motivation for asking this question, is that if the vectors represent word embeddings in natural language processing, then the matrix, should represent a document consisting of a sequence of words ...
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0 answers
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Matrix logarithm of unitary matrix from eigendecomposition of Hermitian matrix

Let $A$ be an $n \times n$ Hermitian matrix and let $A = UDU^*$ be its eigendecomposition. Then what can we say about the Hermitian matrix $B$ where $U = \exp(iB)$? Can we relate it to $A$ in some ...
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1 vote
0 answers
60 views

Matrix logarithm of unitary matrix from polar decomposition of product of positive definite matrices

Let $ \mathcal H = \mathbb{C}^d$ be a finite dimensional vector space and let $X, Y \in \text{Pd}(\mathcal H)$ be two Hermitian positive definite matrices. Let $A = XY$, let $A = UDV^*$ be its SVD, ...
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0 votes
0 answers
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Majorization in restricted set

In Matrix Analysis by Bathia there are statements of the type $x \prec y$ in $\Omega$, where $\Omega$ is some set like $\mathbb R^n$ or $\mathbb R^n_+$. What does it mean? The earliest occurrence I ...
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0 votes
0 answers
34 views

Bound of weighted trace of a self-adjoint matrix

Let $A\in M_n(\mathbb{C})$ be self-adjoint matrix with eigenvalues $\lambda_1\geq\cdots\geq\lambda_n$. Suppose $\{e_j\}_{j=1}^n$ be an orthonormal basis of $\mathbb{C}^n$ and $c_1\geq c_2\geq\cdots \...
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  • 356
1 vote
0 answers
60 views

Minimax type principle for a self-adjoint operator acting on a Hilbert space

Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\left\{\lambda_k(V^*TV):V:\mathbb{C}^k\...
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  • 356
2 votes
1 answer
59 views

Product of unitary matrix and hermitian matrix

I am struggling with this derivation (about wireless channel capacity) while reading a textbook. It seems simple, but I couldn't figure it out. So I am very appreciative if someone explains it to me. ...
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6 votes
1 answer
225 views

Proving an inequality for operators.

Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $$\varphi (A) = -\text {tr}...
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3 votes
0 answers
46 views

Proving that $\varphi$ is operator convex.

Let $\mathbb P_n$ be the space of all $n \times n$ self-adjoint positive definite matrices. Consider the function $\varphi: \mathbb P_n \longrightarrow \mathbb R$ defined by $$\varphi (A) = \text {tr}\...
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0 votes
0 answers
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Multiplying a matrix by a certain unitary matrix: effect on the eigenvalues

Is there any quick way to conclude on the eigenvalues of $T = U S$ ? Can we express them with respect to the eigenvalues of $S$, $\lambda( S)$. Where: $$S = S^T $$ ($S$ is symmetric) $$ U = \begin{...
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0 votes
2 answers
72 views

Prove that $A \mapsto A (A + \lambda)^{-1}$ is Frechet differentiable for any positive matrix $A$ and for any $\lambda \gt 0.$

Show that the map $f : A \mapsto A(A + \lambda)^{-1}$ is Frechet differentiable for any positive matrix $A$ and for any $\lambda \gt 0.$ Also find it's Frechet derivative. Is the space of all ...
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0 votes
1 answer
65 views

How to prove that $\|\Sigma-\mu I\|^2+E\left[\|S-\Sigma\|^2\right] = E\|S-\mu I\|^{2}$?

Let $X$, matrix $n \times p$, with i.i.d. rows, mean $0$ and variance $\Sigma$. The usual unbiased estimator for $\Sigma$ is the sample covariance $$ S = \frac{1}{n} X X^T. $$ Consider the case $p>...
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100 views

The trace property of product of positive operators $\text{Tr}(ABBA)^\alpha = \text{Tr}(BAAB)^\alpha$?

Assume $A,B$ are noncommutative positive operators on $\mathbb{C}^n$, it is easy to see the following trace equality $$\text{Tr}ABBA = \text{Tr}BAAB.$$ Do we have $$\text{Tr}(ABBA)^\alpha = \text{Tr}(...
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1 vote
1 answer
42 views

Matrix Reconstruction from Products

I am currently working on a problem about matrix reconstruction. I am given a theorem and I either need to prove it or disprove it (ideally by a counterexample). The theorem is the following: Let $n,...
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