Questions tagged [matrices]

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 tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

Filter by
Sorted by
Tagged with
3
votes
0answers
61 views

Is every matrix involution over a UFD diagonalisable?

Let $A$ be a UFD, that is also a $k$-algebra, where $k$ is a field of characteristic $\not=2$ (for instance polynomials over $k$). Is every involution in $\mathrm{GL}_n(A)$ diagonalisable? This is of ...
0
votes
0answers
43 views

How to prove the Ky Fan inequality and its opposite

How do I prove that if $A$ and $B$ are Hermitian matrices with eigenvalues $a_1>a_2>\dots >a_n$ and $b_1>b_2>\dots >b_n$ and the eigenvalues of the sum are $c_1>c_2>\dots >...
0
votes
0answers
17 views

Low-discrepancy submatrices in any matrix

First we define what is discrepancy. Discrepancy measure the bias of matrix $M$ and its submatrices. Given a matrix $M$ with entries in $\{0,1\}$, for any combinatorial rectangle R we define $\text{...
0
votes
0answers
13 views

Lagrange Optimization w.r.t. vector + equality constraint

I am currently facing the following optimization problem: $f(v) = r_f + v'*\mu + v'*diag(\Sigma)*1/2 -v'*\Sigma * v*1/2+(1-\gamma)*v'*\Sigma*v*1/2$ s.t. $1'*v=1$ (sum of the vector elements = 1) ...
1
vote
1answer
30 views

Weak majorizations for sum of two hermitian matrices

Let $A$ and $B$ be two $n\times n$ hermitian matrices. Does $U^{*}AU+B \prec_{w} A+B$ for any unitary matrix $U$? Here the notation $``\prec_{w}"$ stands for the weak majorization, that is, $x\prec_{...
4
votes
1answer
166 views

The largest $\ell_p$-norm of a sum of rows of a Sylvester-Hadamard-Walsh matrix

Given any $n\in\mathbb N$, consider the the Sylvester-Hadamard-Walsh matrix $M=(a_{i,j})_{i,j\in 2^n}$ of size $2^n\times 2^n$ and for a number $p\in[1,\infty)$, let $$\nu_{n,p}=\max_{F\subseteq 2^n}\...
2
votes
1answer
63 views

Monotonicity of the determinant of symmetric Toeplitz Matrices

For simplicity, i focus on a particular Toeplitz symmetric matrix, so let $A_{ij} = a^{|i-j|}$ for $i,j=1,\dots,n$ and $0<a<1$ be a Kac-Murdock-Szegő (KMS) matrix, e.g., for n=4 \begin{equation} ...
11
votes
2answers
937 views

The character table of the symmetric group modulo m

Let $S_n$ be the symmetric group and $M_n$ the character table of $S_n$ as a matrix (in some order) for $n \geq 2$. Question: Is it true that the rank of $M_n$ as a matrix modulo $m$ for $m \geq 2$ ...
8
votes
2answers
252 views

$2$-norm of idempotent matrix

Suppose $n > 1$ is an integer. Let $P \in \mathbb C^{n \times n}$ be a matrix such that $P^2=P$ and $1\leqslant {\rm rank}(P)<n$. Prove that $\Vert P \Vert_2 = \Vert I - P \Vert_2$. I have been ...
0
votes
0answers
24 views

Derivative of a hermitian matrix of power $n$ by a scalar [closed]

Suppose I have a parameterized hermitian matrix $H$ whose elements are functions of $x$. Is the derivative of $H^n$ by $x$ equal to $n\frac{d H}{d x}H^{n-1}$, where $\frac{d H}{d x}$ is the ...
3
votes
1answer
80 views

Derivative of an integral of a Gaussian

I'd like to compute the derivative of an expected value w.r.t one of the parameters that define the mean of a Gaussian: $ Z=\int \mathcal{N}(x;\mu,\Sigma)f(x) \, dx $, then $ \frac{dZ}{dK}=\text{??}$ ...
8
votes
1answer
299 views

For every ring R, is there a block-diagonal canonical form for a square matrix over R?

This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily ...
5
votes
1answer
202 views

Combinatorics and symmetry in matrices under row and column swaps

Suppose we have a $m\times n$ matrix and a sequence of numbers with which to fill the matrix, $\{c_1,c_2 \dots c_k \}$. I like to think of the numbers as colors, hence the notation. How many unique ...
3
votes
2answers
277 views

Relation graph isomorphism to discrete logarithm

$\DeclareMathOperator\ora{ora}$Let $A_0$ be the adjacency matrix of graph $G$ and $P_0$ permutation matrix of multiplicative order $\rho$. Let $X$ be positive integer and $B_0=P_0^X A_0 P_0^{-X}$. Q1 ...
2
votes
0answers
53 views

Finding a basis for the range of a linear function

I realize this question is not high level but I have posted it on Math Stackexchange: Stackexchange question and have received some upvotes but no answers or comments, so I am trying here. I will need ...
1
vote
0answers
61 views

Convolution integral and its matrix representation

My background is chemistry and I was exploring some one dimensional deconvolution problems i.e., resolution of two or more overlapping peaks. A lot of excellent work was done in the 1970-80s. However, ...
2
votes
0answers
62 views

Intuition behind eigenvalues of a graph mattering

Is there a good intuition behind why the eigenvalues of a matrix corresponding to a graph tell us useful information about the graph? There are a lot of results relating the eigenvalues of the ...
1
vote
1answer
199 views

Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix: $$ M:= \begin{bmatrix} \cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\ \cPr{X=...
5
votes
0answers
134 views

Matrix decompositions as monoid isomorphisms. Ever considered before?

I've noticed some correspondences between some matrix decompositions and monoid isomorphisms (always to some free commutative monoid), in addition to the one I asked about in a previous question: ...
2
votes
0answers
80 views

The "matrix direct sum" monoid modulo unitary equivalence

Given a commutative $*$-ring $(R,*)$, let $M(R,*)$ be the monoid whose elements are matrices over $R$ of all possible shapes and entries, including those that have $0$ columns or $0$ rows. Let the ...
2
votes
1answer
178 views

Is the number of values the sign function can take on a ring ("signedness") of any fundamental importance? Can it be predicted?

There are well-described methods of generalizing arbitrary functions to matrices in a natural way. Basically, if $A=PD_AP^{-1}$ where $D_A$ is a diagonal matrix, then $f(A)=Pf(D_A)P^{-1}$, where the ...
1
vote
0answers
40 views

Largest eigenvalue of matrix A smaller than 1, what about B when A=B+C? [closed]

Suppose I have a square matrix $A$ that only has non-negative real entries and is not symmetric and not primitive either. It has no "special" structure we could exploit. I know that the ...
5
votes
2answers
469 views

Sorting energy bands in physics

Obtaining the band structure is a standard problem in physics. Supposed there is a system which is described by a Hamiltonian matrix that depends on some system parameter $H(k)$. To find the energies ...
1
vote
0answers
60 views

There is an observation on the eigenvalues of the sum of a kind of special Hermitian matrices. How to prove it?

Suppose $A$ and $B$ belong to a kind of special hermitian matrices, which have the following properties: $A$ and $B$ contain only one negative eigenvalue. the negative eigenvalue and the second-...
0
votes
0answers
102 views

Is the kernel $\vert d_X - d_Y \vert^p$ conditionally negative definite?

Given two finite metric spaces $(X,d_X)$ and $(Y,d_Y)$, for $p > 0$, define the kernel ($4$-D tensor) $K$ on $(X \times Y)^2$ by: $$K\big( (x_i, y_k), (x_j, y_l) \big) = \vert d_X(x_i, x_j) - d_Y(...
0
votes
0answers
65 views

Improving a basis

Given any triple of $n \times n$ matrices $(A, B, J)$ and invertible $2n \times 2n$ matrix $U$ for which $U(J \oplus -J)U^{-1} = \begin{pmatrix}0 & A \\ B & 0\end{pmatrix}$, can we find a pair ...
12
votes
4answers
536 views

Show that the eigenvalues of a non-symmetric matrix built from positive matrices have positive real parts

Let $A, B, C \in \mathbb{R}^{n\times n}$ such that $N = \begin{bmatrix} A & B\\ B^{\top} & C\end{bmatrix}$ is a symmetric positive definite matrix. I'm trying to show that the following matrix ...
0
votes
0answers
78 views

Expressing a diagonal matrix over a division ring

Let $D$ be a division ring and $\mathrm{GL}_n(D)$ the general linear group over $D$. It is not difficult to show that every $A\in\mathrm{GL}_n(D)$ can be written in the form $A=D(x)B$ where $D(x)=\...
1
vote
1answer
140 views

$C\lVert\sum_i a_{ii}\rVert \ge \lVert(a_{ij})\rVert$ for matrices with entries in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\...
1
vote
0answers
48 views

Positive semidefinite fundamental solution to Schrodinger operator

Lets say $V : \mathbb{R}^n \rightarrow \mathbb{M}_d (\mathbb{R})$ is a $d \times d$ symmetric, positive semidefinite matrix function on $\mathbb{R}^n$ and consider the Schrodinger operator $- \Delta + ...
2
votes
0answers
83 views

Product of two involutions in $\mathrm{PSL}_2(D)$

Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two ...
0
votes
1answer
129 views

Does there exist a function $f(X)$ with the following gradient $\mathrm{Tr}[(I-X)^{-1}]\cdot g(X)$?

Let $f: \mathbb{R}^{n\times n} \to \mathbb{R}$ be a function that receives a square matrix and spits out a scalar. Does there exist a function $f$ such that the gradient $\nabla_Xf(X) = \mathrm{Tr}[(...
7
votes
1answer
394 views

Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?

$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$ where $\lVert \rVert$ is the ...
0
votes
1answer
114 views

Product of matrices equal identity

I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$ $$ ((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r, $$ where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...
16
votes
1answer
605 views

Uniquely reconstruct a matrix $M$ from its inverse $M^{-1}$ if $n$ elements of $M^{-1}$ are unknown and $n$ elements of $M$ are given

This question was motivated by a recent MO post. You know $n$ elements of the $N\times N$ matrix $M$ and you do not know $n$ elements of the inverse $M^{-1}$ (but you know the other $N^2-n$ elements ...
7
votes
2answers
264 views

Finding a matrix from its diagonal and the off-diagonal elements of its inverse?

This question comes from https://stats.stackexchange.com/questions/457375/recover-full-covariance-matrix-from-covariance-diagonal-and-precision-off-diagon where it have not found answers. So, let $\...
1
vote
1answer
99 views

Combinatorial graph optimization problem on integer adjacency matrices

We are given a $n\times n$ symmetric matrix $M$ whose entries are positive integers. Let $z_{i,j}:=\frac{M_{i,j}}{M_{i,j}+\sum_{k\neq i,j}\min(M_{i,k},M_{k,j})}$ for all $1\le i<j \le n$, and $z:=\...
2
votes
2answers
212 views

When is $(I - X)^{-\top} \circ X = 0$?

I am currently looking at the following expression $(I - X)^{-\top} \circ X = 0$, where $\circ$ is the Hadamard product, $\top$ is the transpose, $I$ is the identity, and $X$ is non-negative and ...
1
vote
1answer
124 views

Upper bound of rank of a matrix

Given a set $U=\{1,\ldots,c\}$, we have $t$-dimensional vector $v_i=(x_1,x_2,\ldots,x_t)$, where $x_j$ is a subset of $U$. We can check that there are $2^{ct}$ above vectors. Then we construct the ...
1
vote
0answers
123 views

How to fill a rectangle with smaller rectangles of given sizes?

I have a problem. I try to find an algorithm to fill up a given rectangle with smaller ones. Something like in this picture: I know the size of the big rectangle, the size of all the little ...
5
votes
1answer
216 views

Convergence of a series related to $\mathrm{SL}_2({\mathbb N})$

The question The number $\pi$ and summation by $SL(2,\mathbb Z)$ considers the series $$\sum(\lVert x\rVert+\lVert y\rVert-\lVert x+y\rVert)$$ where the series runs over vectors $x,y\in{\mathbb N}^2$ ...
0
votes
1answer
39 views

The eigenvectors and eigenvalues of Laplacian matrix in a chain graph

When I did some research, I have not found the analytical expression about the eigenvalues and eigenvectors of Laplacian matrix in a chain graph while I only found those in a cycle graph. The Laplcian ...
4
votes
2answers
142 views

Computation of the pfaffian of a particular matrix

This question was originally asked in (https://math.stackexchange.com/questions/4265063/computation-of-the-pfaffian-of-a-particular-matrix). I did not find any satisfactory answer there. So I am ...
1
vote
0answers
63 views

Trace minimization for generalized eigenvalue problem

In [1], it is shown in theorem 1.2 that for symmetric $n \times n$ matrices $A$, $B$, we have $$ \min_{Y \in Y^*} \text{tr}(Y^TAY) = \text{tr}(X^TAX) = \sum_{i=1}^p \lambda_i, $$ with $$ \text{ $X^...
1
vote
1answer
80 views

Matrix equation with projection matrix

I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$ $$P - G_1G_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0,$$ where the other quantities are known: $A\in\mathbb{R}^{d\times d}$...
1
vote
0answers
56 views

Derivative of a function of ordered variables

Can I differentiate $$(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)^\top(\pmb{y}^o - (\pmb{x}\cdot\pmb{a})^o)$$ with respect to $\pmb{a}$? (I want to minimize the expression with respect to $\pmb{a}$.) Here, $...
-5
votes
1answer
147 views

Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance, $0^ \left( \begin{array}{cc} \frac{1}{2} &...
2
votes
1answer
122 views

Existence of matrices with some invertibility properties

Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$. I am interested ...
2
votes
0answers
195 views

For a nilpotent matrix A, are the cardinalities of sets: 1) B: commute with A, 2) B: anticommute with A, 3) B: q-commute with A — the same?

Let us work over finite fields $F_{p^k}$. Simulations seems to indicate: Question 1: Consider a nilpotent matrix $A$, consider the set of all matrices $B$, such that $AB-qBA=0$, then cardinality of ...
0
votes
2answers
183 views

Is there a term for the operation of multiplying the product of two matrices by the transpose of the first matrix? [closed]

Is there a term for the operation $A B A^T$? In colloquial terms, I might call this a "sandwich" of a matrix between another matrix and the transpose of that other matrix. How about for the ...

1
2 3 4 5
54