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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.

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Second order matrix PDE

In my research I came across a second-order PDE of the following form $$ \biggl( v \odot \Big( \nabla \rho(x)\,\mathbf{1}^{\top} + ...
Math_Newbie's user avatar
1 vote
1 answer
55 views

Matrix concentration inequality for unbounded (sub-exponential) matrices

Let $a_1, \cdots, a_n\in\mathbb{R}^k$ be independent random vectors sampled from $N(0,\Sigma)$. We aim to establish a high probability bound on the eigenvalues $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...
Nicole's user avatar
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7 votes
1 answer
252 views

Existence of a linear map resulting in the determinant being an elementary symmetric polynomial

Let $1 \leq k \leq n$ be fixed integers. Let $\mathcal{M}_n^{\mathrm{H}}$ be the set of $n \times n$ complex Hermitian matrices (if it makes it easier to answer this question, you may instead use the ...
Nathaniel Johnston's user avatar
6 votes
1 answer
330 views

Questions on symmetric Hadamard matrices

Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal. If $A$ is a symmetric matrix, then $A = A^T$ and if $...
user369335's user avatar
1 vote
0 answers
26 views

explicity quantum expander

A set of $d\times d$ unitary matrices $\{U_1,\cdots,U_n\}$ is called a quantum expander if the identity matrice is the only eigenvector of the linear map $\mathcal{M}(X)=\frac{1}{n} U_i X U_I^{+}$ ...
gondolf's user avatar
  • 1,503
2 votes
1 answer
152 views

Forming real positive semidefinite matrices from complex matrices

I have asked this question on the Mathematics Stack Exchange: https://math.stackexchange.com/questions/4924554/forming-real-symmetric-positive-semidefinite-matrices-from-complex-matrices. Let $Q \in \...
Mthpd's user avatar
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34 views

Efficient compression techniques for low-rank matrix $A$ to maintain multiplication compatibility?

Suppose we have a large low-rank matrix $A$ and an input matrix $X$ (where $X$ is a tall, skinny matrix). We need to compute the product $AX$. To reduce computational cost and data transfer when ...
oleotiger's user avatar
3 votes
1 answer
272 views

Unitary transformations of Vandermonde matrices forms a smooth manifold?

The space of all Vandermonde matrices $V$ with $r$ variables and degree $n$ (as below) forms an embedded submanifold of $\mathbb{R}^{(n+1) \times r}$ when $x_{i} \in \mathbb{R}$. It is naturally a ...
wsz_fantasy's user avatar
1 vote
1 answer
80 views

Where does $V$ from the spectral decomposition $A = VDV^*$ lie, if $A$ has only imaginary entries?

The spectral theorem says that for every Hermitian matrix $A \in \mathbb{C}^{n \times n}$ there is a unitary matrix $V \in U(n)$ and a diagonal matrix $D \in \mathbb{R}^{n \times n}$ such that $A = ...
Tardis's user avatar
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5 votes
1 answer
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What is the "natural" or "physical" norm on the Hessian matrix (and other higher derivatives)?

Let $u : \mathbb R^n \rightarrow \mathbb R$ and let $H : \mathbb R^n \rightarrow \mathbb R^{n \times n}$ be its Hessian matrix. What is the "natural" choice of pointwise norm on the Hessian ...
AlpinistKitten's user avatar
2 votes
0 answers
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Smallest eigenvalue of certain PD matrix decreases under sparse perturbation

Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
PIII's user avatar
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1 vote
0 answers
77 views

Multilinear non-commutative Khintchine inequality

Let $g_1,\ldots,g_k$ be independent standard Gaussians and for each index $(i_1,\ldots,i_k)\in [n]^k$ let $A_{i_1,\ldots,i_k}$ be a $d\times d$ symmetric matrix. Question: Is there a known bound for ...
user293794's user avatar
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Johnson-Lindenstrauss type result for matrix factorization

The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\...
optimal_transport_fan's user avatar
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Given two rectangular matrices and they yield the same results when they are multiplied by their own transposes. What can we say about them?

Suppose we have $MM^T = NN^T$, where $M$ and $N$ are both $n$ by $d$ matrices. Assume that $n$ is (much) larger than $d$, are there anything we could conclude about $M$ and $N$, aside from that $N$ ...
Yrey's user avatar
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Can all real positive semidefinite Hankel matrices be decomposed into sum of rank 1 real positive semidefinite Hankel matrices?

Denote the set of real positive semidefinite $d\times d$ Hankel matrices as $\mathcal{S}$. Can we always decompose one $S\in \mathcal{S}$ into sum of rank $1$ $S_i\in \mathcal{S}$, i.e., $S=\sum_i S_i$...
zzy's user avatar
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1 vote
0 answers
18 views

Correlation Matrix Problem of Three Decomposition Level of DWT

I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix. The step I'm trying to follow is: The DWT coefficientes are obtained from ...
Dragnovith's user avatar
2 votes
1 answer
77 views

Symmetric and anti-symmetric matrices and maximal eigenvalues

Suppose we start with a symmetric $n \times n$ matrix $A$, the elements of which are either $1$ or $0$. All the diagonal elements of this matrix are set to be $0$. So, $\lambda_{\text{max}}=\sup \...
Alapan Das's user avatar
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1 answer
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Matrix quantization and effect on singular values

Let $A$ and $B$ be an $N\times n$ matrix with $n\le N$, and let $\sigma_1(X),\dots \sigma_n(X)$ denote the singular values of $X\in \{A,B\}$. Do we have upper and lower bounds for $$ \| \sigma_i(A)-\...
ABIM's user avatar
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0 answers
34 views

What is the impact of individual estimate on each other in matrix inversion?

I am looking to understand the impact of each estimate on each other in matrix inversion. Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
Sagar's user avatar
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10 votes
3 answers
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Trace inequality for non-reversible Markov chain

Let $P \in \mathbb{R}^{d \times d}$ be the transition kernel for a Markov chain with stationary measure $\pi$ and define $P^\ast$ to be the time-reversed transition kernel defined by $P^\ast_{ij} := ...
Alex Damian's user avatar
6 votes
2 answers
635 views

Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?

I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
bryceadam1's user avatar
-2 votes
1 answer
309 views

Follow-up question re: logarithms of matrix-valued functions known not to have any zero eigenvalues [closed]

Given two parametrised "well-behaved" real skew-symmetric matrix-valued functions (of real variable(s)) $K(x)$ and $L(x)$, is it true that there exists another matrix-valued function $M(x)$ ...
Kanghun Kim's user avatar
8 votes
2 answers
650 views

Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

It is known that an entire function that is nowhere zero must be the exponential of another entire function. Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
Kanghun Kim's user avatar
0 votes
0 answers
103 views

Evaluating a matrix Pick function via its integral representation

In the proof of Theorem 3.1 of the paper Inequalities for M-matrices, Ando evaluates a matrix function (see equation boxed in orange below) via an integral representation of a Pick function (see ...
Pietro Paparella's user avatar
3 votes
1 answer
117 views

Calculate the Riemannian Hessian of Karcher mean problem on positive definite matrices

Consider a collection of positive definite matrices $\{A_1,...,A_n\}\in\mathbb{S}_{++}^d$, the Karcher mean of these matrices is given by (see (5.4) in [1]): $$ \min_{X\in\mathbb{S}_{++}^d} f(X):=\...
Jason Li's user avatar
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12 votes
0 answers
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Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
  • 4,474
1 vote
1 answer
110 views

A sine type Chebyshev system

A sequence of real functions $\{\phi_1,\cdots,\phi_n\}$ is called a Chebyshev system on an interval $I\subseteq\mathbb{R}$, if any real linear combination $\sum_{l=1}^n a_l\phi_l$ has at most $n-1$ ...
ABB's user avatar
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2 votes
0 answers
144 views

What are the name and inverse of an interesting integer matrix?

It is practicable to compute the matrix inverses \begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 2^2 \\ \end{pmatrix}^{-1} &=\begin{pmatrix} 1 & 0 &...
qifeng618's user avatar
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5 votes
2 answers
346 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 ...
li ang Duan's user avatar
1 vote
0 answers
86 views

Positive semidefinite maximum principle

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Let $\mu$ be a Borel probability measure on $M_n(K)$ supported on a compact set $C$ of positive semidefinite matrices with $\mathbf{0}\not\in C$. ...
Joseph Van Name's user avatar
0 votes
0 answers
27 views

How can one orthogonalize the pointwise sum of two orthogonal sets?

Let $n = 2k$, and suppose that $V = \{v_1, \cdots, v_k\}$ is an orthogonal set in $\mathbb{R}^n$. In other words, the vectors in set $V$ are pairwise orthogonal to each other. Now, consider a new set $...
ABB's user avatar
  • 4,030
2 votes
1 answer
187 views

An inequality related to matrix trace

$$Tr(A|A^T U \Sigma U^T|) \leq Tr(AA^T U \Sigma U^T)$$ where $A \in \mathbb{R}^{m\times n}$ is a real rectangular matrix, $U \in \mathbb{R}^{n \times n}$ is a orthonormal matrix and $\Sigma$ is a ...
Wayne's user avatar
  • 21
0 votes
0 answers
19 views

Find condition of X such that I-XSA is nonsingular, where $S$ is skew-symmetric and $A$ is symmetric, nonsingular

Given $I_n$ is the identity matrix, $A \in \mathbb{R}^{n,n}$ is symmetric and nonsingular, and $B\in \mathbb{R}^{n,n}$ is skew-symmetric.\ a) For which condition of a matrix $X \in \mathbb{R}^{n,n}$, ...
IscoBerlin's user avatar
3 votes
0 answers
46 views

Which invertible linear maps preserve the set of Hurwitz stable matrices?

Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
Fiktor's user avatar
  • 1,284
1 vote
2 answers
130 views

Methods to solve for a matrix whose entries satisfy certain properties

(This question is a repost of a deleted question I asked, because the previous version had several elements missing) Setting For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
algebroo's user avatar
  • 113
2 votes
0 answers
49 views

What conclusions can I derive from this family of trace inequalities?

Problem. Let $n_1,\ldots,n_s,m_1,\ldots,m_s\ge 0$ be nonnegative integers and set $m := \sum_{i=1}^s m_i$ and $n := \sum_{i=1}^s n_i$. Let $\oplus$ be an operation on matrices which stacks them in a ...
eepperly16's user avatar
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0 answers
90 views

Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix

Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is $$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\ ...
KAJ226's user avatar
  • 131
4 votes
0 answers
117 views

Matrix product of entire functions

Suppose I have two $d \times d$ entire matrix functions $F, G$ defined on $\mathbb{C}$ with the the property that $\|FG^*\|_{L^\infty(\mathbb{C})} < \infty$. Can anything be said about $F$ and $G$, ...
Joshua Isralowitz's user avatar
8 votes
1 answer
318 views

On a matrix inequality

$\newcommand{\R}{\mathbb R}\newcommand{\tr}{\operatorname{tr}}$It follows from Proposition 7 and this recent answer that, for any positive-definite $n\times n$ symmetric real matrices $A$ and $B$, $$\...
Iosif Pinelis's user avatar
10 votes
2 answers
474 views

Does approximate equality of quantum states imply operator inequality in a large subspace?

Let the trace norm of $X$ be $$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$ and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
Noel's user avatar
  • 165
0 votes
0 answers
81 views

When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?

By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that $$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$ where $e_1,\dots,e_n$ are the standard ...
ABB's user avatar
  • 4,030
4 votes
1 answer
131 views

Sum of holomorphic squares?

Consider a variable $z \in \mathbb{R}^n$ and assume $u(z) \in \mathbb{R}^m$ and $H(z) \in \mathbb{R}^{m \times m}$. Further assume that $H(z)$ is symmetric positive definite for every $z$. Consider ...
Sébastien Loisel's user avatar
2 votes
1 answer
110 views

The eigenvectors of adding a particular rank one matrix to the circulant matrix

Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$. Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
ABB's user avatar
  • 4,030
5 votes
1 answer
300 views

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 ...
user1234567890's user avatar
0 votes
0 answers
140 views

Maximizing the norm of a sum of Hermitian matrices

Consider the following problem: Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
Joseph Van Name's user avatar
1 vote
2 answers
199 views

On a matrix trace inequality

Let $\mathcal H(n)$ be the set of $n\times n$ Hermitian matrices, and $\mathcal S(n) \subset \mathcal H(n)$ be the subset of density matrices, i.e., $A \in \mathcal S(n)$ iff $A$ is Hermitian, ...
Fawen90's user avatar
  • 1,111
3 votes
1 answer
142 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
  • 33
1 vote
1 answer
342 views

Norm inequality

In an article I read, I have the following inequality: $\|A-B\|_1 \geq \max \{ \|A 1_m- B 1_m \|_1, \|A^T 1_n - B^T1_n\|_1 \}$ Where $A, B \in \mathbb{R}_+^{m\times n}$. The $\|\cdot\|_1$ refers ...
CereIssou's user avatar
0 votes
0 answers
92 views

Modeling decay of a linear system with a mixing term

I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence $$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}...
Yaroslav Bulatov's user avatar
4 votes
1 answer
445 views

Positive definiteness of a matrix-valued function

This question is a repost from math.se, where I didn't receive an answer. Are there simple conditions on an $d \times d$ matrix B under which $$ f(t, s) = \begin{cases} \exp(-B |t - s|^\alpha), &...
tsnao's user avatar
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