# 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}
+
...

1
vote

1
answer

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### 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 $\lambda_{\min}(\sum_{i=1}^n a_ia_i^T)$ ...

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answer

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### 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 ...

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votes

1
answer

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### 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 $...

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### 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 $\mathcal{M}(X)=\frac{1}{n} U_i X U_I^{+}$ ...

2
votes

1
answer

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### 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 \...

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answers

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### 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 ...

3
votes

1
answer

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### 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 ...

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vote

1
answer

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### 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 = ...

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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 ...

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votes

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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{...

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### 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 ...

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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\...

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answers

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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 $N$ ...

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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., $S=\sum_i S_i$...

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answers

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### 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 ...

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 \...

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votes

1
answer

79
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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)-\...

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votes

0
answers

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### 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 ...

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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} := ...

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votes

2
answers

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### 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 ...

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votes

1
answer

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### 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 $M(x)$ ...

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votes

2
answers

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### 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 ...

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0
answers

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### 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 ...

3
votes

1
answer

117
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### 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):=\...

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votes

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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 ...

1
vote

1
answer

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### 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 $n-1$ ...

2
votes

0
answers

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### 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 &...

5
votes

2
answers

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### 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 ...

1
vote

0
answers

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### 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 $\mathbf{0}\not\in C$. ...

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votes

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answers

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### 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 $...

2
votes

1
answer

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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 ...

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votes

0
answers

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### 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 $X \in \mathbb{R}^{n,n}$, ...

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votes

0
answers

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### 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 ...

1
vote

2
answers

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### 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 $...

2
votes

0
answers

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### 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 ...

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votes

0
answers

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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 \\
...

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votes

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answers

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### 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 $G$, ...

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votes

1
answer

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### 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 $B$,
$$\...

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votes

2
answers

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### 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 ...

0
votes

0
answers

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### 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 ...

4
votes

1
answer

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### 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 ...

2
votes

1
answer

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### 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 $Te_k=e_{k-1}$ ...

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 ...

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votes

0
answers

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### 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 ...

1
vote

2
answers

199
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### 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, ...

3
votes

1
answer

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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 ...

1
vote

1
answer

342
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### 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 ...

0
votes

0
answers

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### 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}...

4
votes

1
answer

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### 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), &...