Questions tagged [matrix-analysis]

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

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Relating two seemingly different definitions of Riemannian metric.

I've been studying Riemannian geometry from John M. Lee's book on the same. My aim is to understand the geometry of Hermitian positive semidefinite matrices when viewed as a Riemannian manifold. In ...
Afham's user avatar
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Eigendecomposition of a block matrix of the form $\begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$

Consider a matrix $A = \begin{pmatrix} A_{11} & A_{12} \\ A_{12}^T & 0 \end{pmatrix}$ where $A_{11}$ is symmetric. The eigendecomposition of $A_{11}$ is $A_{11} = U_1S_1U_1^T$ and the one of ...
Emma Ceccherini's user avatar
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Backward error of solving $Ax=b$ with perturbed matrix $A$

Let $A=\begin{bmatrix} 1 &-0.55 \\ -2 & 1.06 \end{bmatrix}, b=\begin{bmatrix} 1\\ -1 \end{bmatrix}$. How to compute the backward error of the algorithm of solving $Ax=b$, with respect to ...
HIH's user avatar
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Algorithms to reduce matrix bandwidth

I'm currently working on a variant of a non-convex low-rank matrix completion algorithm, whereby we take a uniform sample of entries in a (symmetric) matrix and look to complete said matrix. For ...
Chandler's user avatar
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What is the second derivative of the spectral norm of a symmetric matrix?

It is well known that the derivative of a matrix $A$'s 2-norm with respect to $A$ is $$\frac{\partial \sigma_{\max}(A)}{\partial A}=u_1v_1^T,$$ where $u_1,v_1$ is the are the first column/row in the ...
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Finding eigenvalues of the sum of two matrices.

If the eigenvalues of two matrices are known is there any (generalized) way to tell the eigenvalues of the sum of the two matrices? I have come to hear that no such theorem exists that can handle the ...
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vector but not matrix norm

Is there a norm $\|\cdot\|_\sharp$ on $\mathbb R^{n\times n}$ for which $\| I\|_\sharp=1$ and there is a matrix $M$ such that $\|M^2\|_\sharp>\|M\|_\sharp^2$?
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Why limit of a matrix's power exits will implies this?

I was reading this paper "Fast Linear Iterations for Distributed Averaging" and on page 3 it reads that: $\lim_{t \to \infty} W^t$ exists (i.e., $W$ is semi-convergent) if and only if there ...
William Zheng's user avatar
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The spectral radius analysis

This is a question from numerical linear algebra. It originates from iteration method: Suppose $Ax = b$, we split $A = A_1+A_2$, then $A_1x = -A_2x+b$, if $A_1$ is invertible, then $x = -A_{1}^{-1}A_2 ...
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Dimension of the manifold of symmetric rank $r$ $n\times n$ matrices

I'm currently reading through the paper "Low-rank matrix completion by Riemannian optimization—extended version" by Vandereycken, and in this paper the author states that the set $\mathcal{M}...
Chandler's user avatar
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Derivatives of Matrix Functions of Different Dimensions

Notation: For matrices $A,B\in\mathbb R^{n\times m}$, we define the inner product $\langle A,B\rangle=\sum_{i,j}A_{ij}B_{ij}$ The basis vector $e_i$ is equal to $1$ at position $i$ and $0$ otherwise....
Blake's user avatar
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Unitary matrix of polar decomposition

I know unitary matrix of polar decomposition ($A=UP$; $U$ is unitary and $P$ is positive semi-definite) cannot be unique (but $P$ is!) if $A$ is not invertible. Can you give some examples that $A=UP=...
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Trace of square of a rank-$1$ Hermitian matrix ${\bf A} = {\bf a}{\bf a}^H$

Given matrix ${\bf A} = {\bf a}{\bf a}^H$, where ${\bf a}$ is a complex column vector. Are the following correct? ${\rm Tr}({\bf A}) = {\rm Tr} \left( {\bf a}{\bf a}^H \right) = {\rm Tr}({\bf a}^H{\...
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Upper bound of biggest singular value of Kronecker Sum via singular values and traces

I am investigating the biggest singular value of Kronecker Sum. I'd like to understand if my argument is correct. Let $A$, $B$ be square matrices of size $n$. Let $I$ be an identity matrix of size $n$....
Piotr Lewandowski's user avatar
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Chain rule for vector by vector derivative

I think it's clear that: \begin{equation} \frac{d(\mathbf{A} \mathbf{x})}{d \mathbf{x}}=\mathbf{A}, \quad \text{ where $\mathbf{A}$ is a matrix and $\mathbf{x}$ is a vector}. \end{equation} but if we ...
Nyquist-er's user avatar
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What will the distribution of $u^HHH^Hu$ when elements of H follows the complex Gaussian independently and u is the eigenvector such that $\|u\|^2=1$?

I tried to solve this problem by directly expanding it. But I need some opinions on my solution. Here is the problem definition: $\lambda=\frac{1}{|| \mathbf{H}^H\mathbf{u}||^2}\mathbf{u}^H\mathbf{H}\...
Joseph's user avatar
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Maximal eigenvalue of Hermitian matrix lies on the diagonal of $A$

Let $A$ be Hermitian. Assume that the eigenvalues of $A$ are increasing ordered and that $a_{ii} = \lambda_n$. We have that $$\sum_{j=1}^{m}a_{ii} \geq \sum_{j=1}^{m}\lambda_j$$ and $$\sum_{j=1}^{m}a_{...
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How to find the Fréchet derivative of a matrix exponential?

Let $f : \Bbb R^{n\times n} \to \Bbb R^{n \times n}$ be the matrix exponential $$ f(A) = \sum_{k = 0}^\infty \frac{A^k}{k!} $$ Let $B$ be a matrix that commutes with $A$. Show that the value of ...
Jimmy's user avatar
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Roots of monic polynomials with real coefficients as continuous functions of coefficients

Consider a monic polynomial $$p(\lambda,a)=\lambda^n+a_1\lambda^{n-1}+\dots+a_{n-1}\lambda^{n-1}+a_n$$ with $a=(a_1,\dots,a_n)\in \mathbb{R}^n$. Suppose that $p(\hat{a},\hat{\lambda}_i)=0$, $i=1,\dots,...
lychtalent's user avatar
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Relationship between matrix 2-norm of internal direct sum and orthogonal vector spaces of traceless matrices

Let A, B be vector spaces of traceless matrices such that A is orthogonal to B (in Frobenius inner product sense) and $S = \{ a + b\ |\ a \in A, b \in B \}$ is an internal direct sum. Is there ...
Piotr Lewandowski's user avatar
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Condition for 3*3 block matrix to be stable

Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...
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How to interpret $A A' = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$

(I edited the question to be more relevant and informative/specific. Hope it's better). I'm unsure of how to interpret $$A A' = \begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$$ where A and A' are ...
MokutekiJ's user avatar
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Convexity of the norm of a matrix exponential

I would like to know the convexity of the function below. Denote the space of $ n \times n $ real symmetric matrices as $\mathcal{S}^n$, then define $f:\mathcal{S}^n \rightarrow \mathbb{R}$ as $$ f(X) ...
gsoldier's user avatar
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Jacobian, vectorization and the Kroenecker product

Suppose $f(U) = U^{T} A U$ with its derivative $d_{f}$ [U] $(H) = H^{T} A U + U^{T} A H$, then its Jacobian is given by $J_f(vec U) = ((AU)^{T} \oplus I) \Pi + I \oplus U^{T} A$, (*) where $\Pi = \Pi^{...
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When does $A_1 A_2 A_3\ldots \xrightarrow{a.s}0$ for IID random matrices $A_i$?

Suppose $A_1,A_2,A_3,\ldots$ is an infinite sequence of $d\times d$ matrices sampled IID from some distribution. Under which conditions does the product converge to zero almost surely? The hard case ...
Yaroslav Bulatov's user avatar
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1 answer
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Can we show the block matrix of the inverse of $A$ is greater than the inverse of the block matrix of $A$?

I know we have the relation $\left( A_{ii} \right) ^{-1}\le \left( A^{-1} \right) _{ii}$ where $A$ is a positive definite matrix. My problem is, can we extend the result to block matrices with $C\...
narip's user avatar
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Bound for generalised Matrix $(p, k)$-norm

Generalised Matrix $(p, k)$-norm[1] generalizes Schatten and Ky-Fan norms as it is defined as follows: $$ N_{p, k}(A) = (\sum_{i=1}^{k}\sigma_i^{p}(A))^{\frac{1}{p}} $$ where $\sigma_i$ is i-th ...
Piotr Lewandowski's user avatar
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Lipschitz continuity of diagonalizing orthogonal matrices

Let $A$, $B$ and $E$ be three symmetric $n \times n$ matrices where $B = A+E$. Let $\mathcal{Q}$ be the set of orthonormal matrices that diagonalize $A$, i.e. the set of orthogonal matrices $Q$ such ...
Daniel De Roux's user avatar
1 vote
2 answers
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Is the derivative of $\pmb x^\top (I+GB)^{-1} (I+GB)^{-\top}\pmb x$ with respect to $B$ a 4th-order tensor?

Is the derivative of $\pmb x^\top (I+GB)^{-\top} (I+GB)^{-1}\pmb x$ with respect to $B$ a 4th-order tensor? Where $\pmb x$ is a vector, and $B$ is a matrix. I followed the procedures in What is the ...
Koji  Tadakoro's user avatar
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ij-element of (A^T)A matrix

I know that given an orthogonal matrix $A\in\mathbb{R}^{n\times n}$, $A^\top A=AA^\top=I_n$. I saw that the $ij$-element of $A^\top A$ can be expressed as $$(A^\top A)_{ij} = (A^\top a_j)_i=(\text{row ...
Jacques's user avatar
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Orthogonal similarity of block real symmetric matrix

Can a block real symmetric matrix whose sum of diagonal blocks is an identity matrix $$\left[ \begin{array}{cccc} H^{T}_{1}H_{1} & H^{T}_{2}H_{2} & \cdots & H^{T}_{1}H_{n+1} \\ H^...
user467247's user avatar
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Differentiation under the integral sign in higher dimensions

Let's say I have a function $f : \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}^n$. Are there conditions under which the following holds $$ \frac{\partial}{\partial\mathbf{x}^T}\int_\Omega f(\mathbf{x},...
RS-Coop's user avatar
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Orthonormal Eigenvectors of $C C^{*}$.

I'm reading a paper, and somewhere in this paper, he considered a set of orthonormal vectors of the matrix $C C^{*}$ corresponding to $1$, where $C$ here is a companion matrix. I tried searching but I ...
Shuichi's user avatar
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2 answers
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If $A^*= A$ and $A^m= 0$, then $A= 0$.

If $A^*= A$ and $A^m= 0$, then $A= 0$. My attempt: If $m = 2$, then $\text{tr}(A^*A)= 0 \Rightarrow A=0$ and then the result follows for $m = \{2,4,8,16,32,\ldots\}$. But I don't know how it works in ...
hemant singh's user avatar
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For any matrices $A$ and $B$ of the same size, show that $\mathrm{Im}(A ,B) = \mathrm{Im}(A) + \mathrm{Im}(B)$.

I just want to be more familiar with block matrices and while I am reading Fuhzen Zhang book, I found this problem $\mathrm{Im}( A,B)$ means the range of the block matrix $$\begin{bmatrix} A & B\...
Farah Palestine's user avatar
1 vote
1 answer
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Is Polar Decomposition Unique?

For an arbitrary complex matrix $A$, it is known that $A$ has either a decomposition $P U$ or $V Q$, where $P = (A A^{*})^{1 / 2}$, $Q = (A^{*} A)^{1 / 2}$, and $U, V$ are isometry. Question: Is it ...
Shuichi's user avatar
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Eigenvalues of Hessian of Lagrange Function

I am interested to know the eigenvalues of the following block matrix, which I obtained from the Lagrange function (optimization). We have $H_{n\times n}$ here positive definite (i.e., all the ...
user602672's user avatar
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preservation of positive semidefiniteness

if P is positive semidefinite, Q is invertible, P and Q are of the same dimension, then is $Q^TPQ$ still positive semidefinite? If so why? edit: I am aware that sylvester's law of inertia might be ...
Sam's user avatar
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Unitary equivalence of two matrices

Let $B\in M_n(\mathbb{C})$ with $BB^*+B^*B=I$. Is $ \begin{pmatrix} 0 & B \\ B & 0 \\ \end{pmatrix} $ is unitarily similar to $ \begin{pmatrix} 0 & \vert B\vert \\ \vert B^*\vert & 0 \\...
Piku's user avatar
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Is $l_2$ on $\mathbb{R}^n$ the only norm for which it is equal to its dual norm?

Given any norm $\|.\|$ on $\mathbb{R}^n$, its dual norm $\|.\|^D$ is defined as the following: $\|v\|^D = \sup_{\|x\|\leq 1} |(v,x)|$, where $(,)$ is the standard Euclidean Inner product. Under that ...
Rohan Didmishe's user avatar
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Finding a real matrix satisfying certain norm requirements

Suppose $M_{100}(\mathbb{R})$ denote the space of $100\times 100$ with real entries matrices with a natural norm. I am trying to show that there exists some $\delta>0$ such that if $M\in M_{100}(\...
neophyte's user avatar
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How to prove that generalized Vandermonde matrix is invertible?

Given $$A = \left( z_i^{\lambda_k}\right)_{i,j = 1,\ldots, n} = \begin{pmatrix} z_1^{\lambda_1} & z_1^{\lambda_2} & \cdots & z_1^{\lambda_n} \\ z_2^{\lambda_1} & z_2^{\lambda_2} & ...
fixingmath's user avatar
3 votes
3 answers
199 views

Interpreting "geometric mean" $\exp(\frac{1}{2} \log(P) + \frac{1}{2} \log(Q))$ for singular symmetric matrices $P, Q$

Let $\mathcal S_+^d$ be set of real $d \times d$ symmetric positive semidefinite matrices and $\mathcal S_{++}^d$ be set of real $d \times d$ symmetric positive definite matrices Following section 1....
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1 vote
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What is the derivative of a matrix-valued composite function?

Preliminaries: Jordan algebra In the sense of Jordan algebra, the following arrow matrix is often used to express the Jordan product $\circ$ which will be defined later: For a vector $a\in\Bbb R^m$, $$...
Keith's user avatar
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Write linear constraints into linear transformation

Suppose we have the following block matrix $$ X=\begin{bmatrix} X_{11} & X_{12} & X_{13}\\ X_{21} & X_{22} & X_{23}\\ X_{31} &X_{32} & X_{33} \end{bmatrix} $$ where $X_{ij} \in ...
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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 ...
Hezaraki 's user avatar
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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+...
ABB's user avatar
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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. ...
Noel's user avatar
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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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