Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level ...

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7
votes
1answer
95 views

Determinant of some covariance matrix (Gaussian kernel process)

Let $x_1,\dots,x_p$ be $p$ points in $\mathbb{R}^n$ ($n\geq 2$) with $x_1=0$. Consider the symmetric matrix $M(x)=(m_{ij}(x))_{1\leq i,j\leq p}$ where $m_{ij}(x) = \exp(-\frac{1}{2}\Vert x_i - ...
2
votes
0answers
52 views

Eigenvectors as continuous functions of matrix - diagonal perturbations

The general question has been treated here, and the response was negative. My question is about more particular perturbations. The counterexamples given in the previous question have variations not ...
-4
votes
0answers
31 views

Large Matrix problem [on hold]

I have been searching the internet for a large matrix problem but didn't find any problem. I am searching for a problem like for example Kirchhoff's Rules Problems where you have I1, I2...etc ...
2
votes
0answers
12 views

Is there some kind of lower bound for estimation error of the estimation of (near) low-rank matrices in high-dimension?

I'm reading S.Negahban and M.J.Wainright's paper, ESTIMATION OF (NEAR) LOW-RANK MATRICES WITH NOISE AND HIGH-DIMENSIONAL SCALING. In the paper, they give a upper bound for estimation error of the ...
-2
votes
0answers
24 views

Matrix decomposition ( Kronecker product decomposition) [on hold]

How to solve the following matrix equation: $Q = A \bigotimes B$ (Given $Q$)? Can we prove that if $Q \in \mathbb{R}^{n^2\times n^2}$ and $Q$ is symmetric, there always exist $A,B$ such that $Q = A ...
6
votes
2answers
189 views

What methods do we have to understand the spectrum of matrices with restricted entries?

Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"), What is the largest possible spectral radius of a $n \times n$ matrix ...
0
votes
1answer
64 views

Reference for measures of commutativity needed

I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C. I'd like a function μ:Cn×n→[0,∞) ...
-2
votes
0answers
54 views

Rotation Matrices [closed]

I am not a mathematician strictly speaking I am in fact a computer science PhD student but my problem couldn't be solved by computer scientist I'm sure. I am actually programming in Android. An ...
2
votes
1answer
362 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
5
votes
2answers
842 views

Matrix derivative with respect to the pseudo-inverse

I'm trying to find a expression for the matrix derivative with respect to the pseudo-inverse of a matrix. So, i have some function $f(A)$ of a matrix $A$, which is singular. If it weren't I could use ...
-4
votes
0answers
44 views

How to solve Ax=b using matlab [closed]

What is the best way to solve XA=B in matlab, where A and B are 3*4 matrices and X is 3*3 matrix.
16
votes
3answers
2k views

Alternate and symmetric matrices

Greetings to all ! Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO. Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is ...
0
votes
4answers
145 views

About adding a negative definite rank-1 matrix to a symmetric matrix

If $B$ is a symmetric matrix then how do its eigenvalues compare to the eigenvalues of $B - vv^T$? ( where $v$ is a vector of the same dimension as $B$) I guess that the eigenvalues of $B - vv^T$ ...
6
votes
1answer
187 views

Convexity of the product of two exponential matrices

Let $S\subset\mathbb{R}$ be a convex set and $\mathbb{S}^{n}$ be the set of real symmetric matrices of order $n\times n$. A matrix valued function $\Gamma: S \rightarrow \mathbb{S}^{n}$ is said to ...
0
votes
0answers
21 views

Orthogonal matrices [migrated]

I've always been told that special orthogonal group (matrices M such as M$M^T$=I and detM=1) represents the rotation. But how do you prove it? I thought it was easy until i tried to do it for ...
1
vote
0answers
52 views

Smallest sum of original column entries in 2d matrix [closed]

I have an interesting optimization problem I am trying to solve now and I thought I'd share it here in order to find the best answer. The problem itself is not complicated and it is stated like this: ...
1
vote
1answer
53 views

Given $M$, minimize $|Mx|_0$

Given a matrix $M \in \mathbb{R}^{n \times m}$, I would like to find $\min_{x \in \mathbb{R}^m} \|Mx\|_0$ such that $x \neq 0^m$, where the $\ell_0$ "norm" is measured by simply counting the number ...
2
votes
1answer
170 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
3
votes
2answers
414 views

S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.
8
votes
2answers
326 views

On closest unitary matrix

In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$. Suppose ...
2
votes
0answers
295 views

How to prove the following determinant identity? [migrated]

This problem is relevant to the spin operator matrix elements in the quantum 1D XY model. For any even integer $N$, define two sets ...
0
votes
0answers
54 views

Derivative of a conjugation of matrices

Let $\mathcal{M}_n$ be the space of complex $n\times n$ matrices. Let $\Phi\colon \mathbb{D}\to \mathcal{M}_n$ and $\psi \colon \mathbb{D}\to \mathcal{M}_n$ be holomorphic functions. Consider the ...
1
vote
0answers
153 views

Is there a way to simplify this apparently huge characteristic polynomial calculation?

Say I am given the $0/1$ adjacency matrix of an undirected graph. Also I am given a representation $\rho$ of some group $G$ and an orientation has been arbitrarily chosen along each edge. Let ...
-1
votes
0answers
19 views

Matrix Calculus and Dot Products [migrated]

I'm a beginner at matrix calculus and I've been using the Wikipedia page for my rules. I'm having some trouble with this problem that I've faced. I know the solution, but I'm wondering why my strict ...
-1
votes
0answers
29 views

Generating pairs of $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ Lie algebras - characterization

It is known that any simple Lie algebra can be generated by two elements (Kuranishi paper). Is it know how to characterize ALL generating pairs for $\mathfrak{su}(N)$ and $\mathfrak{so}(N)$ algebras?
6
votes
4answers
425 views

Analogue of Cayley Hamilton theorem for operators on Hilbert space

Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
0
votes
0answers
61 views

prove that a function is approximatively three dimensional

Let $D_n(x)$ be a diagonal matrix of size $N\times N$ where the $k$th element is $\exp(2\pi\jmath x(n+(k-1)/N)$. Let $P_n$ be a random diagonal $N\times N$ matrix where each diagonal element is a ...
1
vote
1answer
838 views

Conjugate Matrix

Let $A$ be a nilpotent square matrix, $J$ be the antidiagonal matrix with 1's on the secondary diagonal (i.e. $J^{2}=E$) and let $B=J A J.$ Suppose we conjugate the matrices $A,B$ by a matrix ...
4
votes
1answer
140 views

Hankel matrix commuting with a Jacobi matrix

Assume the semi-infinite Hankel matrix $H$ with entries $$H_{i,j}=\alpha_{i+j-1}, \quad i,j\geq1,$$ where $\alpha_{n}\in\mathbb{R}$, is given. It turns out that in some special situations a ...
41
votes
7answers
2k views

How to prove this determinant is positive?

Given the matrices $ A_i= \biggl(\begin{matrix} 0 & B_i \\ B_i^T & 0 \end{matrix} \biggr) $, where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove that $\det(I + ...
2
votes
1answer
145 views

Rank changes with matrix edits

Assume we have rank $r$ real matrix $M\in\{0,1\}^{n\times n}$ (not constrained to symmetric). Assume $W\in\{0,1\}^{n\times n}$ is rank $1$ real matrix. Case $1$: $M+W\in\{0,1\}^{n\times n}$. Could ...
5
votes
0answers
163 views

Characterizing matrices with rank constraint

Given matrix $M\in\Bbb\{0,1\}^{n\times n}$, is there a nice method to characterize $$\mathscr{D}[M]=\{Q\in\Bbb\{0,1\}^{n\times n}:\mathsf{rank}(M-Q)= \mathsf{rank}(Q),\quad M-Q\in\Bbb\{0,1\}^{n\times ...
1
vote
0answers
44 views

Optimization with random matrix

Consider $J$ a random matrix of size $n\times n$ with i.i.d. Gaussian entries $J_{ij} \sim \mathcal{N}(0,\sigma^2/n)$. Let $f(x)=tanh(x)$, and for $x\in\mathbb{R}^n$, $f(x)$ denotes the vector where ...
6
votes
7answers
667 views

Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
4
votes
2answers
988 views

rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$. Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
2
votes
2answers
245 views

Is anything known about the eigenspectrum of the regular representation of the permutation group?

I am looking for information like upper bounds on how many times any eigenvalue can occur or something like how many eigenvalues can be there in some given range. Is anything like this known? The ...
0
votes
0answers
40 views

Centralizer of a non-regular Lie algebra element

It is well understood that the centralizer of a regular element $A$ of a Lie algebra of complex (square, diagonalizable) matrices consists of polynomials $p(A)$ in that element of degree less than $n$ ...
6
votes
1answer
7k views

vector to diagonal matrix [closed]

For any column vector we can easily create a corresponding diagonal matrix, whose elements along the diagonal are the elements of the column vector. Is there a simple way to write this transformation ...
1
vote
0answers
24 views

LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
0
votes
0answers
22 views

Degree $2$ nilpotent matrices with non-zero product [migrated]

Let $n$ be sufficiently large positive integer. Let $M_i$ be $n$ matrices with entries in either $\mathbb{C}$ or $\mathbb{Z}/n \mathbb{Z}$. Is it possible, (1) for $1 \le i \le n, M_i^2=0$ and ...
1
vote
0answers
72 views

Is my particular finite dimension Toeplitz matrix always strictly positive?

Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$. Define a sequence of banded ...
21
votes
4answers
24k views

Eigenvalues of Matrix Sums

Is there a relationship between the eigenvalues of individual matrices and the eigenvalues of their sum? What about the special case when they are Hermitian and positive-definite? I am investigating ...
2
votes
0answers
175 views

An (open?) problem about a sequence of nested principal sub-matrices and their determinants

Problem: Let $A$ be a $n \times n$ integer matrix, $\det(A) = \pm 1$. Under which conditions there exist a nested sequence of principal submatrices of size $n$ such that they all have determinant $\pm ...
2
votes
1answer
223 views

Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent

I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
4
votes
0answers
89 views

Geometric interpretation of the Desnanot-Jacobi Identity

Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the i-th row and j-th column of $M$. The Desnanot-Jacobi Identity states ...
2
votes
1answer
69 views

Does Schatten-p (quasi-)norm satisfy the norm inequality for 0<p<1?

I'm reading the paper by ANGELIKA ROHDE AND ALEXANDRE B. TSYBAKOV, ESTIMATION OF HIGH-DIMENSIONAL LOW-RANK MATRICES. And in the paper, they provide an inequation of the Schatten-p (quasi-)norm, ...
7
votes
0answers
79 views

What is the symmetry group fixing norms of elements of a unitary matrix?

Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$. When exactly are two unitary matrices related in this ...
1
vote
2answers
126 views

Norm of a matrix operator with a special structure

Let $\{\alpha_{n}\}_{n\in\mathbb{N}}$ be positive sequence such that $$\sum_{n=1}^{\infty}\alpha_n<\infty.$$ Question: Is there any chance to evaluate the operator norm of the matrix operator ...
25
votes
1answer
760 views

Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$

Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle. Assume that the eigenvalues ​​of $A$ are included in a circle arc of ...
6
votes
3answers
302 views

Is $1/\max(i,j)$ a bounded matrix on Hilbert spaces?

I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$. It would be sufficient to know if the Lehmer matrix ...