Questions tagged [matrix-analysis]
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
715
questions
6
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1
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Best rank one approximation
Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$
\begin{equation*}
\arg\min_{x\in\mathbb{C}^n} \|...
6
votes
0
answers
714
views
Spectral norm bound on smooth primary matrix function perturbation
Consider an $L$-Lipschitz function $f: \mathbb{R} \to \mathbb{R}$ (so $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$) and Hermitian PSD matrices $A, B \in \mathbb{C}^{n\times n}$. Define $f(A)$ to be $f$ ...
6
votes
0
answers
306
views
Can we approximate any eigenvalue of an infinite matrix via eigenvalues of some sequence of submatrices which approximates the matrix?
Let $T:\ell^2\to\ell^2$ be a compact linear operator. Let $[T]=(a_{i,j})_{i,j=1}^{\infty}$ be the representing infinite matrix of $T$ with respect to the canonical base. Let $T_n$ be the finite rank ...
6
votes
0
answers
563
views
Lower bound on the sum of singular values for a sum of Hermitian matrices
Denote the eigenvalues of an $n\times n$ matrix $\mathbf{X}$ by $\lambda_i(\mathbf{X})$ and its singular values by $\sigma_i(\mathbf{X})$, $i=1,\ldots,n$. When $\mathbf{X}$ is Hermitian, we know that $...
6
votes
0
answers
238
views
Operator arithmetic-harmonic mean inequality with operator-valued weights
Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
6
votes
0
answers
475
views
Symmetric matrices with $\rho(A)\gg\|A\|_\infty$
For a symmetric real matrix $A$, denote by $\rho(A)$ the spectral radius of $A$, and by $\sigma(A)$ the largest absolute row sum of $A$; that is, $\sigma(A)=\max_i \sum_j |a_{ij}|$, where $a_{ij}$ are ...
6
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0
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When is the inverse diagonally dominant?
There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and ...
5
votes
3
answers
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How to check whether a matrix is completely positive or not?
The definition:
cone of completely positive matrices
$$
\mathcal{C}=\left\{
\sum_{i=1}^kx_ix_i^T : \text{$x_i\in\mathbb{R}^n_+$ for $i=1,2,\ldots,k$}
\right\}.
$$
I just don't know how to check ...
5
votes
1
answer
651
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Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
5
votes
1
answer
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A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
My question pertains to this paper by Terence Tao and Van Vu, https://arxiv.org/abs/math/0703307
Both my questions pertain to the argument presented in this paper in its section 6 (page 5). We are ...
5
votes
2
answers
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Nontrivial lower bound on the sum of matrix norms
Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is
\begin{equation}
\|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2.
\end{...
5
votes
2
answers
759
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Optimization version of the Sylvester equation
The Sylvester equation is a matrix equation of the form $AX-XB=C,$ where $A,B,C$ are given matrices of dimension $m\times m,n\times n$ and $m\times n$ and $X$ is an unknown matrix of dimension $m\...
5
votes
5
answers
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Spectrum of transition matrix for symmetric random walk
I asked this question previously on math.stackexchange.com, where it had little traction.
Consider the symmetric random walk on $\{0,1,…,n\}$ with transition probabilities $P(j→j±1)=1/2$ for $0 &...
5
votes
1
answer
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Diagonalization of symmetric matrices of functions
I asked this question some time ago in MSE but I didn't recieved any feedback.
https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions
This problem arised ...
5
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2
answers
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Finding roots of a recursively given polynomial sequence / eigenvalues of an almost Toeplitz matix
I would like to find the roots of the polynomial sequence given by a recurrence relation as follows:
$V_0(x) = 1-a^2$
$V_1(x) = 1-a^2 - x$
$V_{k \geq 2}(x) = (1+a^2 - x)V_{k-1}(x) - a^2V_{k-2}(x)$
...
5
votes
1
answer
783
views
A matrix norm inequality II
Let $\|\cdot\|$ be the spectral norm, i.e., largest singular value. The condition number of an invertible complex matrix $X$ is defined as $\kappa(X):=\|X\|\|X^{-1}\|$.
I am able to prove
...
5
votes
2
answers
342
views
How expressive is $e^A$ in the sense of universal approximation?
For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
5
votes
1
answer
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Analytical form for the nuclear norm of an $n \times n$ matrix
I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
5
votes
2
answers
208
views
Distance of low-rank matrices to the identity for the $\infty$-norm
I am trying to get a lower bound (or even the exact value) of
$$
\min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m
$$
where $m \leq n$, and the ...
5
votes
1
answer
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Is a pointwise " simple tensor" valued continuous map a tensor product of two continuous maps?
A matrix $A\in M_{4}(\mathbb{C})$ is called a simple tensor if $A=B\otimes C$ for two $2\times 2$ matrices $B,C$.
Assume that $X$ is a Hausdorff topological space.Assume that $f:X\to M_{4}(\...
5
votes
2
answers
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How to calculate the square root of matrix $A+B$ perturbatively?
$A=diag\{\lambda_1,...,\lambda_n\}$ and $\lambda_i>0$, $B$ is a positive definite symmetric matrix and $max\{B_{ij} \}\ll min\{\lambda_i\}$
Note that the perturbative calculation of square root ...
5
votes
1
answer
373
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Best orthogonal approximation of rank 1 matrix
Let $X=\lambda_0u_0v_0^T\in\mathbb{R}^{n\times n}$ be a rank 1 matrix where $\lambda_0\in\mathbb{R}$, $u_0,v_0$ are of unit Euclidean norm. What is the solution of the following problem?
$$\hat{X}=\...
5
votes
1
answer
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Inverse of a matrix and the inverse of its diagonals
While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have
$${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D}...
5
votes
1
answer
389
views
Trace of a nonlinear matrix equation
Let $X_0$ be a trace-one positive definite matrix, i.e. $X_0>0$, $\mathrm{tr}(X_0)=1$. Let $A>0$ and consider the following iteration
$$
X_{k+1} = X_k^{1/2}AX_k^{1/2},\quad k\geq 0,\quad (\star)
...
5
votes
1
answer
389
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Matrix integrals in combinatorics, for dummies
This is actually about one particular question that I posted a while ago, "Special" meanders. Among several approaches tried is a huge subclass of approaches which can be generated from ...
5
votes
3
answers
669
views
Norm of triangular truncation operator on rank deficient matrices
Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ A\|\le\frac{\...
5
votes
2
answers
464
views
Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices
I'll first provide the background.
Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables
$x^{(1)},\ldots,x^{(n)}$.
We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
5
votes
2
answers
684
views
Maximal norm-1 projection
Suppose I have a real unitary matrix $U$ and a unit vector $\mathbf{x}, \|\mathbf{x}\|_2 = 1$. What is the solution to the following problem?
$$
\widehat{\mathbf{x}} = \arg\max_{\mathbf{x}, ~\|\...
5
votes
3
answers
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Spectral properties of the LDL^T matrix factorization
Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...
5
votes
1
answer
226
views
A Schur-like product theorem on groups
Let $G$ be a finite group, and consider the composition $X * Y$ on $\mathbb{C}G$ defined by $$(\sum_g \alpha_g u_g) * (\sum_g \beta_g u_g) = \sum_g \alpha_g \beta_g u_g.$$ This composition can be ...
5
votes
1
answer
133
views
connected stable rank
There is a beautiful formula by Leonid Vaserstein relating the Bass and topological stable rank of a commutative unital Banach algebra A to
that of the matrix algebra M_n(A). Is there something ...
5
votes
1
answer
722
views
Proving a majorization inequality for the singular value of the product of two matrices without using tensor product
For any two matrices $\mathbf{A},\mathbf{B} \in \mathbb{C}^{n \times n}$, we know that the following majorization inequality holds
$$
\tag{1}
\label{grz}
\sigma^{\downarrow}(\mathbf{A}\mathbf{B}) \...
5
votes
1
answer
728
views
On faces of convex sets of positive semidefinite matrices
A face $F$ of a convex set $C$ is a set such that, if $x \in F$ is a convex combination of other elements in $C$, then they must also be in $F$. I will denote by $F(C,x)$ the smallest face of $C$ ...
5
votes
1
answer
266
views
Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?
I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...
5
votes
1
answer
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Upper bound of the largest eigenvalue of a PSD block matrix in terms of blocks
Let $\mathbf A=\left[\begin{matrix}\mathbf A_{11}&\mathbf A_{12}\\ \mathbf A_{21}&\mathbf A_{22}\end{matrix}\right]$ be a positive semi-definite matrix, $\mathbf A_{ij}\in\mathbb C^{n\times n}...
5
votes
1
answer
335
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Log determinant of quadratic form
I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide ...
5
votes
2
answers
304
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Is this subset of matrices contractible inside the space of non-conformal matrices?
Set $\mathcal{F}:=\{ A \in \text{SL}_2(\mathbb{R}) \, | \, Ae_1 \in \operatorname{span}(e_1) \, \, \text{ and } \, \, A \, \text{ is not conformal} \,\}$, and
$\mathcal{NC}:=\{ A \in M_2(\mathbb{R}) \...
5
votes
2
answers
958
views
Lower bound for smallest eigenvalue of symmetric doubly-stochastic Metropolis-Hasting transition matrix
For my master's thesis research, I stumbled upon a question concerning the Metropolis-Hasting transition matrix $W$.
Context $\quad$
Let me start with some context. I consider connected undirected ...
5
votes
1
answer
685
views
Optimization of a function of a positive definite matrix and its inverse
This question is a little ill-posed, but I've been playing with some equations and am just wondering if this resembles any known problems that have been solved.
Suppose I have two real, positive ...
5
votes
1
answer
257
views
Epidemic threshold
Need some help / ideas to proceed. Stuck for a while on this.
In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{\max}(A)$ where $\lambda_{\max}(A)$ is the ...
5
votes
2
answers
423
views
Simultaneous maximization of two Generalized Rayleigh Ritz Ratios
Consider hermitian positive semi-definite matrices $A_1$ and $A_2$. Consider also positive definite matrices $B_1$ and $B_2$. I want to maximize the minimum of the two Generalized Rayleigh Ritz ratios ...
5
votes
2
answers
552
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Perron Frobenius with one negative pair of entries
Suppose you have a real symmetric matrix $A$ which is positive except for $a_{ij},a_{ji}$, who are negative.
While it is not generally true that the eigenvector of the dominant eigenvalue of $A$ is ...
5
votes
1
answer
313
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Is the set $\{\zeta: \rho(A(\zeta))< 1\}$ connected for matrices under parameterization of first $m$ rows?
$\newcommand{\eqqcolon}{=\mathrel{\vcenter{:}}}$
Fix $n, m \in \mathbb N$ with $n > m$. Let $\zeta \in \mathcal{M}(m \times n; \mathbb C)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \...
5
votes
1
answer
378
views
Simultaneous Tridiagonalization of a given set of hermitian matrices?
I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?
5
votes
1
answer
283
views
Are there any known results on numerical ranges of rank-one positive semi-definite matrices?
In my problem, I came across numerical ranges of rank-one positive semidefinite matrices. Through Toeplitz-Hausdorff theorem and some other extensions, I know if there are at most three matrices, then ...
5
votes
0
answers
261
views
Analysis proof of dual number spectral theorem
Let the dual numbers be $\mathbb R[\varepsilon]/(\varepsilon^2)$. Write a general dual number as $a + b \varepsilon$ (where $\varepsilon^2 = 0$). Given a symmetric matrix $M$ over the dual numbers (i....
5
votes
0
answers
683
views
Gershgorin's 2nd theorem (disjoint circles): elementary proof?
Let $A \in \mathbb{C}^{n\times n}$ be a complex matrix. We let $a_{i,j}$ be the $\left(i,j\right)$-th entry of $A$ for all $i, j \in \left[n\right]$ (where $\left[n\right]$ denotes $\left\{1,2,\ldots,...
5
votes
0
answers
89
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Schur norm of weighted Cauchy matrix
The Schur norm of a matrix $A$ is defined to be $\|A\|_S=\max\{\|A\circ X\|: \|X\|\leq 1\}$, where $\|\cdot \|$ is the operator norm of a matrix, i.e., the largest singular value.
Let $a_1,\ldots, ...
5
votes
0
answers
558
views
A minimal eigenvalue inequality
Suppose $A$ is an $n\times n$ real symmetric positive definite matrix. Let $A^{(-1)}_{i,j}$ be the $n\times n$ matrix the entries $(i,i),\,(i,j),\,(j,i),\,(j,j)$ of which equal to the corresponding ...
5
votes
0
answers
411
views
When is a Hankel matrix invertible?
I would like to have some conditions that render a general Hankel matrix of the form
\begin{pmatrix}a_1 & a_2 & a_3 &\cdots & a_n \\ a_2 & a_3 & a_4 & \cdots & a_{n+1} \...