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Questions tagged [matrix-analysis]

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

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Forming real symmetric positive semidefinite matrices from complex matrices.

Let $Q \in \mathbb{C}^{n\times n}$ be any matrix. When can we say that the matrix $A=Q^{t}Q$ (where $Q^{t}$ denotes the transpose of a matrix) is a real symmetric positive semidefinite matrix? Write $...
Mthpd's user avatar
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How to find range subset statements for 2x2 block matrices

I am currently dealing with 2x2 block matrices. In my quest of understanding some basics, I discovered the following Lemma: Let $A \geq 0$. Given a subspace $S$, partition $A$ as before: \begin{...
Bruno's user avatar
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Schatten p-norm inequality $\|A-B\|_p ≥ \|D_A-D_B\|_p$

I am seeking a reference or proof for the following. Given two Hermitian matrices $A$ and $B$, and their corresponding diagonal matrices $D_A$ and $D_B$ containing their eigenvalues, I am interested ...
Dante Perès 's user avatar
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1 answer
26 views

The monotonically increasing range of a matrix norm function

I would like to find the monotonically increasing range of a function related to matrix norms: $$ f(x)=\left\|I-e^{-Ax}\right\| $$ Where $I$ is the identity matrix, $A\in\mathbb{R}^{n\times n}$ is a ...
Nap Tsirk's user avatar
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1 answer
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Distance between subspaces with spectral norm

I was trying to prove this following theorem , Let $$ W=\begin{bmatrix} \underset{\scriptscriptstyle n\times k} {W_1} && \underset{\scriptscriptstyle n\times (n-k)} {W_2} \end{bmatrix} $$ $...
Kaustubh Limaye's user avatar
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If a singular square matrix is multiplied by a non-singular square matrix, the null space of the result is what?

If a non-singular square matrix $B$ is multiplied by a singular square matrix $A$ of the same order, the nullspace of the resulting matrix $C=B\times A$ or $C'=A \times B$ remains unchanged from that ...
X.H. Yue's user avatar
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Does every positive semidefinite hankel matrix obeys one Vandermonde decomposition?

I'm reading the paper(I can't find one arXiv version of this paper...) and suspect the correctness of one theorem inside. A hankel matrix $H$ is a square matrix in which each ascending skew-diagonal ...
narip's user avatar
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Finding the eigenvalues of a tridiagonal block matrix of special form

Consider the following symmetric tridiagonal block matrix: $$\begin{bmatrix} 2I_{N \times N} & -I_{N \times N} & O & \dots & O &O \\ -I_{N \times N} & 2I_{N \times N} &...
Stack_Underflow's user avatar
2 votes
1 answer
90 views

On the logarithm of a matrix

While teaching a course on ODE, I needed to introduce the notion of matrix logarithms. I intend to define it as follows. Definition (Matrix Logarithm) Let $A\in GL_n(\mathbb{C})$. We define A) ...
Tatin's user avatar
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Generalization of Sylvester's law of inertia to the case of rectangle matrix

Sylvester's law of inertia states given a symmetric matrix $A$ and a squared invertible matrix $S$ of the same size, then $A$ and $SAS^\top$ have the same number of positive, negative, and zero ...
Simon's user avatar
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The lower bound of operator norm

The operator norm of a mapping $\mathcal{A}:\mathbb{R}^{n_1\times n_2} \to \mathbb{R}^{n_1\times n_2}$ is defined by $$ \|\mathcal{A}\|=\sup_{\|X\|_{F} \leq 1}\|\mathcal{A}(X)\|_{F}, $$ where $X$ is a ...
Kim's user avatar
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Is the form $P_{1}(P_{1} + P_{2})^{-1}$ a contraction elementwisely where both $P_{1}$ and $P_{2}$ are positive definite

In $\mathbb{R}^{n}$, assume $P_{1}$ and $P_{2}$ are both positive definite. Let $A = P_{1}(P_{1} + P_{2})^{-1}$. Can we conclude the following, Given every vector $x$, we have \begin{equation*} |(Ax)_{...
hmeng's user avatar
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How to show this matrix limit?

Let $A$ be a $m\times n(m<n)$ matrix, $\mu>0$. $A$ is a full rank matrix. I am interested in showing that $$\lim_{\mu\to 0} (A^TA+\mu I)^{-1}A^T= A^T(AA^T)^{-1},$$ where $I$ is the identity ...
Du Xin's user avatar
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Simplifying Summation of Outer Products

Let $X \in \mathbb{R}^{d \times n}$ and $Y \in \mathbb{R}^{d \times m}$ have columns $x_{i}, y_{j} \in \mathbb{R}^{d}$ respectively. Is there any way to write the matrix $$\sum_{i = 1}^{n}\sum_{j = 1}^...
slitherhiss's user avatar
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Dense subset of $GL(n)$ and existence of elements with some properties.

In the following $|.|$ denotes the operator norm in the space $GL(n):=$set of invertible matrix of size $n \times n$. I have the next problem: Let $S \in GL(n)$ and $\mathcal{G}$ be a dense subset of $...
C L 's user avatar
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On estimating $\exp(-iHt)$ when $H$ is perturbed

Let us assume that we have an unbounded Hamiltonian $H$ and we perturb it a bit to be $H'=H+ \varepsilon A$. I am sure that estimating $||\exp(-iHt) - \exp(-iH't)||$ belongs to the subject of ...
Lwins's user avatar
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Operator Monotony of composed maps

It is a common fact from Matrix Analysis (see e.g. Bhatia 1997, "Matrix Analysis") that the map $t\mapsto t^r$ is matrix and in fact operator monotone on $t\in[0,\infty)$ if and only if $r\...
JanK's user avatar
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How to prove this variation-of-constants formula?

If $A ,B\in \mathbb{R}^{n\times n},v\in\mathbb{R}^n$,then from the variation-of-constants formula $$\exp((A+B)\tau)v=\exp(A\tau)v+\int_0^\tau \exp(As)B\exp((A+B)(\tau-s))vds$$. In my opinion,this ...
Du Xin's user avatar
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51 views

Relations between row and columns of orthogonal matrix

Given a orthogonal matrix $Q \in \mathbb{R}^{n\times n}$, we know $Q^{\top}$ is also orthogonal. Let $Q$ represent a linear transformation from an Euclidean space to itself, then reading from the ...
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Matrix inequality involving square root and positive matrices

Let $A$ and $B$ be positive-semidefinite matrices with norm satisfying $\|A\| \leq 1$ and $\|B\| \leq 1$. Is the following inequality true? $\|\mathbb 1 -AB-\sqrt{\mathbb 1-A^2}\sqrt{\mathbb 1-B^2}\| \...
Dante Perès 's user avatar
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1 answer
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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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40 views

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 $u_1,v_1$ are the first column/row in the SVD ...
William Zheng's user avatar
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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 ...
TshrD23's user avatar
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1 answer
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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 $\|M^2\|_\sharp>\|M\|_\sharp^2$?
Oskar Limka's user avatar
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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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1 answer
102 views

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 ...
Stack_Underflow's user avatar
3 votes
1 answer
144 views

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
2 votes
1 answer
47 views

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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64 views

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=...
user avatar
1 vote
0 answers
72 views

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{\...
BoltzBooz's user avatar
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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
1 vote
1 answer
282 views

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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22 views

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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1 vote
1 answer
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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_{...
WHERE 234's user avatar
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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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1 vote
1 answer
65 views

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
1 vote
0 answers
35 views

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 ...
Zishuo's user avatar
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1 answer
112 views

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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2 votes
2 answers
110 views

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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0 answers
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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
2 votes
1 answer
288 views

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
  • 67
2 votes
1 answer
92 views

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
0 votes
1 answer
45 views

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
37 views

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
1 vote
1 answer
134 views

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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1 vote
0 answers
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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
0 votes
1 answer
346 views

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},...
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