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About $CW(512,16^2)$

Definitions: A weighing matrix $W = W(n,k)$ with weight $k$ is a square matrix of order $n$ and entries $w_{ij}$ in $\{0, \pm 1\}$ such that $WW^T=kI$, where $I$ is the identity matrix. A circulant ...
user369335's user avatar
6 votes
0 answers
111 views

Factorization to sparse matrices

$\newcommand{\lrank}{\operatorname{lrank}}$ $\newcommand{\rank}{\operatorname{rank}}$ Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it. Now, given ...
Daniel Weber's user avatar
  • 3,319
1 vote
0 answers
63 views

Reference request for non-banded Toeplitz matrix

I want to know references that treat the property of eigenvalues and eigenstates of the non-banded Toeplitz matrix. I mean for example, the Toeplitz matrix $A$ whose matrix element is given by $A_{ij}=...
hos's user avatar
  • 11
6 votes
1 answer
239 views

Attempts to define a matrix exponential over (as much as possible) general fields

Given a $n \times n$ matrix $A$ over the complex numbers, the exponential of $A$ is defined as $$\exp(A) := \sum_{k = 1}^\infty \frac1{k!} A^k , \qquad \tag{$\star$}\label{468645_star}$$ where ...
rosan98's user avatar
  • 361
5 votes
1 answer
103 views

Interpolation between two matrices so that $L^p$ norm is controlled

Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
tommy1996q's user avatar
0 votes
1 answer
524 views

What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
Olórin's user avatar
  • 179
5 votes
1 answer
141 views

On the half-skew-centrosymmetric Hadamard matrices

Definition 1: A Hadamard matrix is an $n\times n$ matrix $H$ whose entries are either $1$ or $-1$ and whose rows are mutually orthogonal. Definition 2: A matrix $A$ is half-skew-centrosymmetric if ...
user369335's user avatar
8 votes
1 answer
323 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
5 votes
1 answer
474 views

An inequality for certain positive-semidefinite matrices

Suppose that $G=(G_{ij})$ is a positive-semidefinite symmetric matrix with the diagonal entries all equal $1$ and all off-diagonal entries $\le0$. Does it then necessarily follow that $$\sum_{i,j}(G^5)...
Iosif Pinelis's user avatar
3 votes
1 answer
203 views

Results on Boolean matrices

Matrices with entries in the finite field of two elements $\mathbb{F}_2$, and with the usual operations of matrix addition and multiplication, have been intensively studied, especially due to their ...
goll-y's user avatar
  • 31
7 votes
0 answers
195 views

Hölder continuity of spectrum of matrices

Endow $\mathbb{C}^{d \times d}$ with the norm induced by the Euclidean norm on $\mathbb{C}^d$. It is well-known (to those who know it well, I guess) that the spectrum $\sigma(A)$ of a matrix $A \in \...
Jochen Glueck's user avatar
2 votes
0 answers
233 views

Do you know this formula for the scalar product in barycentric coordinates?

I've found a formula for a scalar product in barycentric coordinates which I think is pretty cool. I hope that it's new. Is it? Suppose that you have points $x_1,\dots,x_n$ sitting in general position ...
Vladimir Zolotov's user avatar
2 votes
1 answer
460 views

Maximum permuted row/column sum of a matrix

Given a real $n \times n$ matrix $A$ (feel free to assume its entries are non-negative, if it helps), what is known about the problem of computing the quantity $$ \max_{\sigma \in S_n} \left\{\sum_{j=...
Nathaniel Johnston's user avatar
20 votes
2 answers
1k views

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
jack's user avatar
  • 3,153
6 votes
0 answers
392 views

Divisibility properties of minors of matrices

Let $A$ be an $m\times n$ matrix with integer entries. Let $d_i(A)$ be the greatest common divisor of all $i\times i$ minors of $A$, and define $d_0(A)=1$. Whenever $i\leq j$, one has that $d_i(A)$ ...
Joel Louwsma's user avatar
1 vote
0 answers
83 views

Properties of a matrix built via a "matricization" of a unit vector [closed]

Suppose I have a unit vector $\vec v$, and I write it as a matrix, e.g., $16$-vector $\vec v=(v_1,\dots,v_{16})$, where $v_i$ is the $i$-th entry of the vector $\vec v$, is written as follows $$\begin{...
narip's user avatar
  • 119
3 votes
0 answers
147 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, ...
ACR's user avatar
  • 879
3 votes
1 answer
156 views

How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?

Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$. I would like to show that $$ \{ \mathbf{x} \in \mathbb{A}^n_{\...
Johnny T.'s user avatar
  • 3,625
4 votes
3 answers
369 views

Determinant in terms of certain $2\times 2$ minors

Let $A$ be an $n\times n$ matrix with entries $a_{i,j}$. Define an $(n-1)\times(n-1)$ matrix $B$ with entries $b_{i,j}=a_{1,1}a_{i+1,j+1}-a_{1,j+1}a_{i+1,1}$. Then $\det(B)=a_{1,1}^{n-2}\det(A)$. I ...
Joel Louwsma's user avatar
2 votes
1 answer
741 views

rank of a linear combination of matrices

Let $A_1,..., A_s \in M_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that $$ m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\}}...
Johnny T.'s user avatar
  • 3,625
2 votes
0 answers
99 views

When does a matrix subspace contain a full rank matrix?

Cross-posted at Math SE Let $S\subseteq M_{n,m}(\mathbb{C})$ be a $d$-dimensional subspace of the space of $n\times m$ complex matrices (with $n\leq m$, say). I am interested in figuring out ...
mathwizard's user avatar
3 votes
0 answers
148 views

Spectrum of symmetric Toeplitz matrix

A matrix is Toeplitz if it is constant on the diagonals parallel to the main diagonal. I am looking for references on the spectrum of finite symmetric Toeplitz matrices over finite fields.
Patrick Sole's user avatar
2 votes
0 answers
130 views

Pfaffian generalization

The identity $$\left| \begin{array}{cccc} x & y_1 & y_2 & y_3 \\ z_1 & 0 & a & b \\ z_2 & -a & 0 & c \\ z_3 & -b & -c & 0 \\ \end{array} \right|=\...
Alexey Ustinov's user avatar
13 votes
0 answers
237 views

A Dynkin type classification result in linear algebra

Let $G$ be a finite directed acyclic graph. The Cartan matrix $C_G=C$ of $G$ is defined as the matrix with rows and colums indexed by the vertices of $G$ and $c_{i,j}$ counts the number of paths from $...
Mare's user avatar
  • 26.5k
9 votes
1 answer
472 views

$M = AA^t$ where $A$ has unit norm columns

Let $M \in \mathbb{R}^{k\times k}$ positive definite with $\operatorname{tr} M = m$, where $m$ is an integer such that $m \geq k$. I have found a way (using this answer) to decompose $M = AA^t$ with $...
Yair Daon's user avatar
  • 185
4 votes
0 answers
1k views

Reference for matrices with all eigenvalues 1 or -1

In a homological algebra problem I am in the situation that I have an invertible (over $\mathbb{Z}$) integer matrix $X$ and a permutation matrix $Y$ such that $N:=XY$ is a matrix with all eigenvalues ...
Mare's user avatar
  • 26.5k
3 votes
1 answer
5k views

Relation between the eigenvalues of a block matrix and the eigenvalues of its diagonal blocks

Consider the $(m+n) \times (m+n)$ block matrix $$M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ I need references where they are talking about the relation between the eigenvalues of $M$ ...
GA316's user avatar
  • 1,269
1 vote
1 answer
254 views

When does $\det \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} = (\det A)^2 + (\det X)^2$?

Let $A$ be an $n \times n$ real symmetric matrix. Let $$ M = \begin{pmatrix} A & X \\ X^T & A \end{pmatrix} $$ where $X$ is a real invertible $n \times n$ matrix. I am interested in finding ...
Johnny T.'s user avatar
  • 3,625
11 votes
1 answer
1k views

A square root inequality for symmetric matrices?

In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
leo monsaingeon's user avatar
3 votes
0 answers
39 views

A non-singularity property for sets of real matrices

Let $M_N(\mathbb{R})$ be the ring of $N\times N$ real matrices. We say that a couple $(\mathcal{U},\mathcal{V})$, with $\mathcal{U},\mathcal{V}\subseteq M_N(\mathbb{R})$ is admissible if, for every $A\...
Capublanca's user avatar
44 votes
2 answers
2k views

Fermat's Last Theorem for integer matrices

Some years ago I was asked by a friend if Fermat's Last Theorem was true for matrices. It is pretty easy to convince oneself that it is not the case, and in fact the following statement occurs ...
Luis Ferroni's user avatar
  • 1,889
8 votes
3 answers
663 views

Representation theorem for matrices (reference request)

Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that $$ A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k, $$ where $\lambda_1,\dots,\lambda_n$...
Jochen Glueck's user avatar
3 votes
0 answers
180 views

Automorphisms of infinite matrix algebra

This is a similar question to one that I posted in MSE a few days ago. I recently came across this paper from Alahmedi, Alsulami, Jain and Zelmanov, which quoted the following result for $M_\infty(K)$...
dbossaller's user avatar
3 votes
0 answers
359 views

Do we know what the impulse to "introduce" the Jordan canonical form was?

Mo-ers, Do you know how it was that the study of the Jordan canonical form began? There are certain things that may be said once one has thought about the matter: for instance, one can say that the ...
Jamai-Con's user avatar
3 votes
0 answers
122 views

Algebra of block matrices with scalar diagonals

I am interested in block matrices $A$, that is $A\in M_{n\times n}(R)$ where $R=M_{s\times s}(k)$ and $k$ is a field, such that for every positive integer $m$ the matrix $A^m$ has only scalar blocks ...
Adam Przeździecki's user avatar
5 votes
1 answer
514 views

Do matrices with only elements along the main and anti-diagonals have a name?

To expand upon the title, I am wondering if there is a specific name for square matrices of the form: $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} ...
Victoria M's user avatar
1 vote
0 answers
158 views

Non-trivial ways for generating matrices $A$ for which $A + A^T$ is positive-definite?

Disclaimer: This might be an SE question, but I'm not quite sure... Thanks in advance! Setup So, it is known (see Proposition 5.2) that if $A + A^T$ is positive-definite then $A$ must be a $P$-...
dohmatob's user avatar
  • 6,853
4 votes
2 answers
1k views

Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....
5 votes
2 answers
249 views

Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as $$ A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...
Ludwig's user avatar
  • 2,712
1 vote
0 answers
619 views

On the basis of a finite dimensional vector space (revised)

Revision in response to the comments to earlier version: To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
hinkali's user avatar
  • 51
4 votes
1 answer
413 views

A Handbook of Matrix Factorizations

I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
Fixed Point's user avatar
2 votes
1 answer
217 views

Diagonalising a symmetric matrix with polynomial entries

Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
Johnny T.'s user avatar
  • 3,625
11 votes
1 answer
632 views

How do computer algebra packages like Sagemath implement rank of a matrix

I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work. I have been using ...
Nikhil's user avatar
  • 263
5 votes
1 answer
315 views

Rank-constrained least-squares solution of the Sylvester matrix equation

For the Sylvester matrix equation $AX+XB=C$, I want to find the least-squares solution $X$ via $$\begin{array}{ll} \text{minimize} & \| AX + XB - C \|_{\text{F}}^2\\ \text{subject to} & \mbox{...
dave2d's user avatar
  • 191
14 votes
4 answers
3k views

Vandermonde matrix is totally positive

A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
Loïc Teyssier's user avatar
4 votes
0 answers
244 views

On the sum of the first row of the inverse of a certain symmetric Toeplitz matrix

(i) Consider a Toeplitz matrix $A_n = (a_{i, j})_{1 \le i, j \le n}$ of size $n$ defined as follows: $$ a_{i, j} := |i-j|^{-1/2}, \text{ if } i \ne j; \ \ a_{i, j} := 2, \text{ if }i = j. $$ Let $...
Kazuki OKAMURA's user avatar
2 votes
0 answers
102 views

Eigenvalue distribution for a real-valued random matrix with correlated Gaussian entries

I'm working on an application where I would greatly benefit from knowing the distributions of the eigenvalues of a real-valued random matrix whose elements can be assumed to be Gaussian, but where I ...
Ian Cero's user avatar
  • 121
7 votes
3 answers
1k views

Determinant of correlation matrix of autoregressive model

I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form $$R = \begin{pmatrix} 1 & r & \cdots & r^{d-...
Nikolayevich's user avatar
1 vote
0 answers
47 views

Partial ordering of a matrix entries [closed]

I need this for experimentation in some work, so it is not without purpose. Consider the in-spiraling and out-spiraling $4\times 4$ matrices $$\begin{pmatrix} 1&2&3&4 \\ 12&13&14&...
T. Amdeberhan's user avatar
1 vote
1 answer
111 views

Decomposition of integral non-generate matrices [closed]

Let $GL_{\eta}(n,\mathbb{Z})=\left\{a\in GL(n,\mathbb{R})\cap M^{n\times n}(\mathbb{Z})|det(a)=\eta\right\}$. Prove that there exists a finite number of matrices $a_i$ in $GL_{\eta}(n,\mathbb{Z})$ ...
Markiff's user avatar
  • 333