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
0 votes
0 answers
100 views

Trace of matrices and cyclotomic fields

Motivated by Myerson's comment, I have found a way to simplify the question. Let $p$ be a prime, $n\in\mathbb{Z}^+$ not divisible by $p$, and $C$ an $n\times n$ diagonal matrix over $\mathbb{C}$ with $...
user avatar
  • 135
2 votes
0 answers
118 views

Eigenvalues of the sum of matrices, where matrices are tensor products of Pauli matrices

recently I've been studying the toric code (a squared lattice in the context of quantum computation). I want to calculate the energy of the ground state and of all the excitations, with the respective ...
user avatar
  • 21
2 votes
1 answer
57 views

conjugacy in adjoint representation

Let $G$ be an adjoint algebraic group over $\mathbb{C}$, $\mathfrak{g}$ its Lie algebra. Let $\rho:G\rightarrow GL(\mathfrak{g})$ be the adjoint representation. Let $g,g'\in G$ be two semisimple ...
user avatar
  • 3,173
0 votes
1 answer
53 views

Full matrix ring over an infinite division ring with a finite maximal unital subring?

I'm wondering if there is an infinite division ring $D$ and a finite unital subring $R$ of the full matrix ring $M_n(D)$ ($n$ some positive integer) such that there are no rings properly between $R$ ...
user avatar
  • 9
3 votes
0 answers
33 views

Invertibility of the sampling matrix

Given a function $f: \mathbb{R}^2\rightarrow\mathbb{C}$ sampled as a matrix $F_{ij}$ on some ractangle $[a,b]\times[c,d]\subset\mathbb{R}^2$ with steps $\Delta x$ and $\Delta y$ as the stepsizes so ...
user avatar
  • 71
3 votes
0 answers
78 views

Construct a special kind of SVD

Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as $$ A = XD_AY^H \\ B = XD_BY^T $$ where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
user avatar
  • 71
1 vote
0 answers
80 views

Construct special "joint SVD" from separate SVDs

Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as $$ A = XD_AY^T \\ B = XD_BY^T $$ where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
user avatar
  • 71
5 votes
3 answers
233 views

Finite index subgroup of $\mathbb{Z}^2$ that is invariant under a non-singular matrix

Let $M $ be a matrix in $ \operatorname{GL}(2, \mathbb{Z})$ that has at least one eigenvalue of absolute value strictly bigger than $1$. What are the finite index subgroups $H$ of $\mathbb{Z}^2$ such ...
user avatar
  • 233
2 votes
1 answer
193 views

What is the minimum number of multiplications for $2\times 3$ and $3\times 2$ multiplication?

Strassen demonstrated a seven multiplication algorithm for $2\times 2$ matrix multiplication and Winograd showed its optimality. Let $A$ be $2\times k$ and $B$ be $k\times 2$. What is the minimum ...
user avatar
  • 12.8k
4 votes
1 answer
210 views

How to get $\lim_{N\to \infty} \sum_{i=1}^N e^{\lambda_i}u_i^2=\int e^{\lambda}d\sigma(\lambda)$?

I am reading the one lecture note Dynamics for Spherical Models of Spin-Glass and Aging. On page 126. In the Sherrington-Kirkpatrick (SK) model, we suppose that there are $N$ people labeled as $[N]:=\{...
user avatar
  • 156
2 votes
1 answer
144 views

What is the name of a matrix operation using the OR operator instead of addition?

Let's say we have two matrices $M$ and $G$ with $G, M \in \{0, 1\}^{n, n}$, we denote by $m_{i, j}$ the element of $M$ in the $i^\text{th}$ row and $j^\text{th}$ column, same for $G_{i, j}$. Let's ...
user avatar
  • 31
2 votes
1 answer
90 views

An upper bound on an invertible matrix

I have looked through books such as Matrix Analysis by R.A. Horn and C.R. Johnson and would not find an answer to the following question: Given $V^TV \in S^{n}$, where $V$ is an invertible matrix with ...
user avatar
  • 23
1 vote
0 answers
65 views

Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms

I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations. Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
user avatar
0 votes
0 answers
25 views

What are the convergence requirements for Inverse Power Method?

I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below. ...
user avatar
1 vote
0 answers
50 views

Eigenvalues of two positive-definite Toeplitz matrices

Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are: $$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
user avatar
3 votes
2 answers
140 views

Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration

I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
user avatar
12 votes
4 answers
3k views

Why is fast matrix multiplication impractical?

I am wondering why fast matrix multiplications are impractical, especially for Boolean matrix multiplication. I read some content saying fast matrix multiplications are impractical because of large ...
user avatar
0 votes
0 answers
75 views

Adding the AWGN to the data makes its covariance matrix always positive definite?

I'm working on a numerical method that estimates direction-of-arrivals in antenna arrays. I realized that every time I add the AWGN (Additive white Gaussian noise) to a data (which is a matrix), its (...
user avatar
0 votes
0 answers
73 views

Classification of elements $GL(d, \mathbb{R})$

Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here. Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
user avatar
  • 761
0 votes
0 answers
28 views

Integration of matrix form of Vasicek variance (Python/Matlab)

$X_t$ is a vector and follows the following Vasicek process. $$ dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\ $$ What is the variance of $X_t$? In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
user avatar
  • 1
2 votes
0 answers
38 views

spilt the sum of singular values of matrices

Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
user avatar
  • 761
15 votes
4 answers
881 views

Possible values of the determinant for matrices with elements $\{1, 0, -1\}$

For matrices with elements $\{-1, 1\}$ it is known from here that the possible absolute values of determinants of $n \times n$, $n \leq 6$ matrices with entries $\{-1, 1\}$ are as follows: ...
user avatar
15 votes
2 answers
1k views

Vanishing of a sum of roots of unity

In my answer to this question, there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity $$\sum_{k=0}^{N-1}z^{2k^2+k}$$ ...
user avatar
  • 47.1k
1 vote
0 answers
71 views

Determinant Inequality with unitary matrix

I come up with the following conjecture while doing my research, which is a determinant inequality. I have tried to run the MatLab simulation to verify its sanity. It seems that the inequality is true....
user avatar
  • 95
14 votes
3 answers
1k views

Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?

The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
user avatar
  • 275
-2 votes
1 answer
98 views

Proving 2 matrices have the same trace [closed]

I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is: Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...
user avatar
3 votes
2 answers
116 views

Iterative methods for linear system with non-diagonally dominant matrix

I have a linear system \begin{align*} \left[\begin{array}{cccc} 1 & 2 & 1 & -1 \\ 3 & 2 & 4 & 4 \\ 4 & 4 & 3 & 4 \\ 2 & 0 &...
user avatar
  • 65
0 votes
1 answer
77 views

Correct way to conduct equilibrium scaling of linear/integer/MIP program

I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ...
user avatar
  • 21
1 vote
0 answers
26 views

Factorization of an invertible matrix $A$ into $A=MB$ with $M$ symplectic

A matrix $M \in \mathbb{R}^{2n\times 2n}$ is called symplectic if $M^TJM=I$ and $J$ is the standard symplectic matrix $$ J = \begin{pmatrix} 0 & -I \\ I & 0 \end{pmatrix} \in \mathbb{R}^{2n\...
user avatar
7 votes
1 answer
274 views

All eigenvalues are nonnegative

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \mathrm{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \mathrm{M}_n ( \mathbb{Z} )$ such that $...
user avatar
  • 2,767
7 votes
2 answers
377 views

When is the rank of $AB+BA$ equal to one?

For two arbitrary matrices $A$ and $B$, are there any known conditions for the rank of $AB+BA$ to be equal to one?
user avatar
2 votes
1 answer
101 views

Why does the $k$-th invariant factor of some matrix depend only on the first $k$ columns

Let $A$ be an $n\times n$ integral matrix and $\xi$ an integral vector of dimension $n$. Let $$W_i=[\xi,A\xi,\ldots,A^{i-1}\xi]$$ for each positive $i$. Let $d_{i,j}$ (where $1\le j\le i$) denote the $...
user avatar
  • 375
8 votes
1 answer
383 views

Expected rank of linear combination of matrices

Let $A_1,\dots, A_s \in M_n (\mathbb{R})$ 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 \{...
user avatar
-1 votes
1 answer
130 views

Public key cryptography based on non-invertible matrices, part II

Closely related to this question and extending comment of R. van Dobben de Bruyn. Working over $\mathbb{F}_p$ and all matrices of square $n \times n$. Alice chooses invertible $X_A$ and non-...
user avatar
  • 23.3k
1 vote
1 answer
75 views

Diagonalizing a symmetric block matrix

Let us consider the matrix $$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$ then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$ Now, let us consider a block matrix $$ A = \begin{...
user avatar
  • 19
5 votes
1 answer
514 views

Public key cryptography based on non-invertible matrices?

Added Wed 13 Apr 2022 I have written a short note with experimental data, which shows not all pseudo keys are good keys. Public key cryptography based on non-invertible matrices We got public key ...
user avatar
  • 23.3k
4 votes
1 answer
108 views

Maximal eigenvalue of a correlation matrix with some entries fixed as zeros

Let $A$ be real a positive semidefinite matrix of dimension $n$ and with $1$s on the diagonal. Those matrices are sometimes referred to as correlation matrices. From the positivity of the minors, we ...
user avatar
3 votes
2 answers
133 views

Solving linear matrix equation

Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
user avatar
1 vote
0 answers
44 views

Equivalence constants for induced matrix norms

Disclaimer: I asked this question beforehand on mathematics stack exchange, but I think it is better suited for this site Given two sets $P_i\in\mathbb{R}^s$, bounded, convex, with non-empty interior ...
user avatar
  • 119
6 votes
0 answers
76 views

Hölder inequality inside trace

$\DeclareMathOperator\tr{tr}$Suppose we have positive semidefinite matrices $A_1, \dotsc, A_n$ and $B_1, \dotsc, B_n$ of the same dimension. Do we have a Hölder inequality for the trace of the ...
user avatar
  • 411
0 votes
1 answer
37 views

How sparse can a matrix mapping between sparse vectors be?

Let $s$ and $d$ be non-negative integers with $0\leq s<d$ and let $v,u\in \mathbb{R}^d$ be vectors satisfying the sparsity estimate $$ \max\{\|u\|_0,\|v\|_0\}\leq d-s, $$ where, as usual, for any ...
user avatar
4 votes
0 answers
139 views

Number of {0,1}-matrices with an even number of 1’s in each row vs in each column

I am working on an equation that would be solved if I show the following. Let $k \geq l$, and consider the set of $\{0,1\}$-matrices of size $k \times l$ with exactly $i$ 1’s. Consider the subset $\...
user avatar
  • 643
1 vote
1 answer
170 views

Monotonicity of eigenvalues II

In a previous question here, I asked the question below for block matrices and received an answer showing the question is true if $\mathcal B$ is hermitian and false, in general if $\mathcal B$ is non-...
user avatar
  • 417
6 votes
1 answer
360 views

Monotonicity of eigenvalues

We consider block matrices $$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and $$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$ Then we define the new matrix $...
user avatar
  • 417
1 vote
1 answer
81 views

Distribution of weight of special type of random-matrix vector product?

Let $G$ be a matrix of dimension $k \times n$ sampled uniformly randomly from $F_2^{k \times n}$. It is a well known fact that $y = xG$ is uniformly distributed in $F_2^n - \{0\}$ for all $x \in F_2^k$...
user avatar
6 votes
1 answer
145 views

Writing upper triangular 0-1 matrices as a product of a permutation matrix and an upper triangular matrix

Let $C$ be an upper triangular matrix with entries 0 or 1 such that every diagonal entry is equal to one. Let $M_C:=-C^{-1}C^T$. Question: Is there a nice direct criterion (or even classification) on ...
user avatar
  • 21.8k
15 votes
1 answer
721 views

Conjugated subgroups in $\mathsf{GL}(m+n,\mathbb{Z})$ implies conjugated subgroups in $\mathsf{GL}(n,\mathbb{Z})$?

In my research I came up with the following question: Question: Let $H_1$ and $H_2$ be finite abelian subgroups of $\mathsf{GL}(n,\mathbb{Z})$. Define $$ H_1'=\left\{\begin{pmatrix} I_m &0\\0&...
user avatar
8 votes
1 answer
222 views

Cardinality of the maximum points of the determinant on matrices with entries in [-1, 1]

By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many ...
user avatar
  • 665
6 votes
1 answer
195 views

Vanishing linear combinations of minors

Let $V$ be the set of $k$ by $n$ matrices ($k<n$) with entries in $\mathbb{C}$, and let $\mathbb{C}[V]$ denote the set of polynomial functions on $V$. For any subset $I \subseteq [n] = \{1,2,\dotsc,...
user avatar
  • 135
2 votes
1 answer
179 views

Effect of duplicated row on singular values and vectors

Let $\mathbf{A}$ be a $n\times n$ matrix with Singular Value Decomposition (SVD) $\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{V}$ and $\mathbf{a}_1$ be the first row of $\mathbf{A}$. What can we say about ...
user avatar
  • 129

1
2 3 4 5
56