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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.

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14 views

Smallest eigenvalue for Gram matrix of unit norm matrices

Given $n$ symmetric matrices $A_1, \dots, A_n \in \mathbb{R}^{k\times k}$, such that $\|A_i\| \leq 1$ for all $i$, we consider the matrix $M \in \mathbb{R}^{n\times n}$, where $M_{ij} = \langle A_i, ...
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0answers
16 views

Infinite spectral norm of linear mapping

Suppose we have a linear mapping $A:\mathbb{R}^m\rightarrow \mathbb{R}^n$. We define its $k$-spectral norm as: $\sigma_k(A)=\sup_{x} \frac{||Ax||_k}{||x||_k}$. We know that when $k=2$, $\sigma_2(A)$ ...
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0answers
29 views

On the exponent of a certain matrix $A$ in characteristic $p > 0$

Let $A$ be a square matrix in characteristic $p > 0$ with both column and row having length $(1 + p + \cdots + p^i)$, where $i > 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ of the ...
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votes
1answer
58 views

Finding the minimum sum of a subset of entries of a given matrix with combinatorial constraints

Given a matrix $M\in\mathbb{N}^{n\times n}$, let $Z$ be the set of all the $M$'s entry subsets $S$ such that (i) no two entries of $S$ are on the same row or column of $M$ and (ii) $|S|=n$. Clearly we ...
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0answers
52 views

inequality for the difference between largest eigenvalues of an irreducible matrix $A$ and a matrix $B$ with all the entries non-negative

I found an inequality for the difference between largest eigenvalues of an irreducible matrix $A$ and a matrix $B$ with all the entries non-negative (with the assumption that its largest eigenvalue is ...
2
votes
0answers
130 views

Schemes obtained replacing variables by $n \times n$ matrices?

Let $k$ be an algebraically closed field and $n \geq 1$ be an integer. Choose a polynomial $f \in k[x_1,\dots,x_m]$ and place the variables in a fixed order (i.e we choose a preimage of $f$ inside $k\...
3
votes
2answers
118 views

Completely positive matrix with positive eigenvalue

A matrix $A \in \mathbb{R}^{n \times n}$ is called completely positive if there exists an entrywise nonnegative matrix $B \in \mathbb{R}^{n \times r}$ such that $A = BB^{T}$. All eigenvalues of $A$ ...
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0answers
45 views

How to find a “near LCM” of multiple numbers or common multiples with the least emainder that isn't zero [closed]

I'm not sure if I am explaining this right but basically I am trying to figure out a "near LCM". If I have 9 pears, 3 apples and 4 bananas the LCM is 36. But there is a "near LCM" in 9 where 3x3 is ...
5
votes
2answers
183 views

Are there large integer matrices with entries computable in polynomial time, such that all minors are nonzero?

Is there a sequence of matrices $(A_n\in M_{2^n\times2^n}(\mathbb{Z}))_{n\in\mathbb{N}}$ such that the $(i,j)$th entry of $A_n$ is computable in polynomial time, such that all minors of each $A_n$ are ...
8
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1answer
342 views

A question about special linear group

Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying $$ AGA^{-1} \in SO(n,\mathbb R)? $$
8
votes
1answer
121 views

Distance between subalgebras and positive elements in matrices

I repost here from stackexchange, as I was not given an answer there. (https://math.stackexchange.com/questions/2956530/distance-between-subalgebras-and-commutants-in-matrix-algebras) This is a ...
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0answers
47 views

Matrix eigenvalues inequality (3)

Assume that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a_1\ge a_2\ldots \ge a_n>0;$ $B$ is a $m \times m$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_m>0;$ $...
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0answers
90 views

Matrix eigenvalues inequality (2)

Suppose that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a_1\ge a_2\ldots \ge a_n>0;$ $B$ is a $m \times m$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_m>0;$ ...
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0answers
33 views

Marginal Distribution of Partition Matrix

Suppose that $X\sim IW_{p}(n,I_p)$ has an inverse Wishart distribution, which probability density function is $$f(X\mid n)\propto |X|^{-\frac{n+p+1}{2}}exp\Big(-\frac{1}{2}tr( X^{-1})\Big),~~\qquad (...
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0answers
29 views

Reachability of eigenvalues for additive matrix equation

Suppose I am considering the following matrix-valued equation: \begin{align} A = B + C,\\ \text{where } A,B,C \in \mathbb{R}^{n\times n} \end{align} My aim is: Given $B$, I want to find a ...
2
votes
1answer
328 views

Matrix eigenvalues inequality (1)

Assume that $A$ is a $n\times n$ positive matrix, whose eigenvalues are $a_1\ge a_2\ldots \ge a_n>0;$ $B$ is a $p \times p$ positive matrix, whose eigenvalues are $b_1\ge b_2\ldots \ge b_p>0.$ ...
0
votes
0answers
31 views

random matrix integration

Assume $V$ and $W$ are both random positive matrix, $a,b$ and $c$ are real constant. Question: What is the sufficient and necessary conditions of a,b,c for the following integration is integrable? $$...
0
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0answers
32 views

Stability analysis with minimal spectral norm

Let $A \in \mathbb{R}^{n \times n}$ with $$ s(A) = \inf_{D \textrm{ is diagonal}} \| D^{-1} A D \|_2 > 1 $$ Does there exists $m \in \mathbb{N}^n$ and $z \in \mathbb{C}$ with $|z| > 1$ such ...
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1answer
48 views

Intersection between a line and a n dimensional parallelotope

Suppose that I have a line in an $n$ dimensional space described by $$ X=A+Bk, \quad \quad X,A,B \in \mathbb{R}^n, k \in \mathbb{R} $$ here $A$ is known and I want to find all the possible vectors $B$ ...
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0answers
54 views

An inequality concerning the eigenvalues and eigenvectors of an SPD matrix

Let $Ax_i=\lambda_ix_i, \ (i=1,\cdots,n)$ be an eigensystem of the symmetric positive-definite diagonally-dominant matrix $A=\{a_{ij}\}$. Let $$b_{jk}=\sum_{i=1}^{n}{\frac{(x_i(j)-x_i(k))^2}{\...
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0answers
24 views

Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set \begin{equation} S_{\{1,2\}} := \{...
26
votes
1answer
540 views

Determinants of binary matrices

I was surprised to find that most $3\times 3$ matrices with entries in $\{0,1\}$ have determinant $0$ or $\pm 1$. There are only six out of 512 matrices with a different determinant (three with $2$ ...
12
votes
1answer
301 views

Are there nice isomorphisms $\operatorname{S}^2(k^n)\cong\Lambda^2(k^{n+1})$?

This might be forced to migrate to math.SE but let me still risk it. The spaces $\operatorname{S}^2(k^n)$ and $\Lambda^2(k^{n+1})$ from the title have equal dimensions. Is there a natural isomorphism ...
2
votes
1answer
185 views

Are Linear Maps resistant to Noise?

Let's assume I have a $m \times m$ matrix $M$ with Frobenius norm $1$ and a unit vector $x \in S^{m-1}$. I also have a second $m \times m$ matrix $M^*$ which is obtained from the first one plus some ...
10
votes
0answers
192 views

What are these matrices called?

A paper I'm writing heavily uses block diagonal matrices with the property that each block is upper triangular and constant along its diagonal. Like this: $$\begin{bmatrix} A_1 \\ & \ddots \\ &...
3
votes
0answers
68 views

How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$? [duplicate]

Suppose I have two matrices $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$. I want to know what's the best similarity transformation between these two matrices when we ...
3
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0answers
47 views

How to show that a continuous family of symmetric matrices is uniformly positive?

My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$: $ \{A(\lambda,x_1,x_2) ; (x_1,...
8
votes
2answers
263 views

Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$

Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
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0answers
64 views

Similar reduced integral matrices

Let $d\geq 1$ be an integer. I'm not assuming anything here about $d$ (it is not necessarily squarefree, for example) For all $a,b,c\in\mathbb{Z}$ such that $ac-b^2=d,$ set $[a,b,c]_d=\begin{pmatrix}...
2
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2answers
53 views

Lower bound of positive entropies of automorphisms on tori

Let $A$ be an automorphism on tori $\mathbb{T}^d$. It is well known that the topological entropy $$ h(A)=\sum_{\lambda} \max\{0, \log|\lambda| \} $$ where $\lambda$ goes through all eigenvalue of $A$ ...
3
votes
1answer
162 views

Volume of polyhedron

Given the following polyhedron: All the $n\times n$ matrices $\boldsymbol{X}$ with elements $x_{ij}\in(0,1)$ such that $$\boldsymbol{X}\cdot\boldsymbol{1}=\boldsymbol{r}, \boldsymbol{1}^T\boldsymbol{...
10
votes
1answer
251 views

Automorphisms of $GL_n(\mathbb{Z})$

I want to consider the crossed module: $H \xrightarrow{t} Aut(H)$ for the case where $H = GL_n(\mathbb{Z}) = Aut(T^n)$ is the automorphism group of the $n$-torus. Any suggestions on how to understand ...
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1answer
52 views

Solving Problem: LMIs and block matrices

I have been reading through this paper (https://ieeexplore.ieee.org/document/7995739) where I am stuck with this particular LMI. If you are familiar with control theory, the author is trying to find ...
5
votes
1answer
284 views

$(AB)^+\approx B^+A^+$ for $B$ “fat” enough?

Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse. Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
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0answers
86 views

Operator norm of a soft thresholded symmetric matrix

Let $A$ be a symmetric real-valued $n\times n$ matrix and let ${\left\|A\right\|_{2\rightarrow 2}} := \max_{\left\|u\right\|_{2}\leq 1} \left\|Au\right\|_{2}$ denotes its operator norm (largest ...
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0answers
37 views

Exact eigendecomposition of a specific Toeplitz matrix

I am interested in diagonalizing a general $n \times n$ matrix with entries of the form \begin{equation} \frac{1}{|f_i-f_j|^p} \hspace{40px} 1 \le i,j \le n \end{equation} where $f_i,p \in \mathbb{R}$ ...
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0answers
35 views

Eigenstructure and condition number of a block symmetric matrix

Consider a block, symmetric matrix $$ \begin{pmatrix} 0 & -A^T \\ A & C \end{pmatrix} $$ where $A$ and $C$ are two real positive definite matrices. What is the condition number of that ...
4
votes
1answer
113 views

A generalization of invariant and coinvariant subspaces

Let $\mathcal{A}$ be a subalgebra of $M_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for ...
2
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0answers
55 views

Orthogonal Matrices and Cosets (translates) of Linear Subspaces

Let $M_n(F_2)$ be the vector space of all $n\times n$ matrices over the finite field $F_2$. Let $O(n)\subset M_n(F_2)$ be the set of all orthogonal matrices and $W\subseteq O(n)$ be an affine subspace ...
7
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0answers
113 views

A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...
4
votes
1answer
130 views

The minimum rank of a matrix with a given pattern of zeros

For real matrices $A=(a_{ij})$ and $B=(b_{ij})$ of the same size, I write $A\prec B$ if $a_{ij}=0$ whenever $b_{ij}=0$. If $$ B = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & ...
0
votes
1answer
51 views

Estimating Maximal-Clique of Metric Graphs via the Rank of their Adjacency Matrix

Let $\mathrm{M}\in\lbrace0,1\rbrace^n$ be the adjacency matrix of a graph $\mathrm{G}\left(V,E\subseteq\lbrace\lbrace u,v\rbrace| u,v\in V\rbrace\right)$ of order $n$. Let $\mathrm{G}$ ...
15
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0answers
234 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
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0answers
20 views

Matrix Product Chain Representation of an Addition Chain [closed]

I am looking for references to anything interesting thats know about matrix product chains that take the vector $\{1\}$ to another vector $\{n\}$ (the end result of an addition chain). Each matrix ...
2
votes
0answers
49 views

On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...
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votes
0answers
33 views

A $\mathbb C$-linear map from $M(p-1,\mathbb C)$ to $\mathbb C^\hat G$, where $p$ is an odd prime and $G=\mathbb Z/(p) ^\times$

Let $p$ be an odd prime and $G=(\mathbb Z/(p))^\times=\{1,2,...,p-1\}$ i.e. $G$ is a cyclic group of order $p-1$. Let $\hat G:=\{\chi:G \to \mathbb C^\times : \chi $ is a group homomorphism $\}$. For ...
26
votes
0answers
349 views

Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
2
votes
0answers
70 views

k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
3
votes
1answer
78 views

Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...
1
vote
1answer
104 views

Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller. Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...