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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.

2
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2answers
51 views

Rate of convergence for eigendecomposition

Consider the discrete Dirichlet Laplacian on a set of cardinality $n.$ For example the Dirichlet Laplacian $\Delta_D$ on a set of cardinaltity 4 is the matrix $$\Delta_D := \left( \begin{matrix} 2 &...
2
votes
1answer
53 views

Bounding entries of the inverse of a matrix with bounded entries

Let $A$ be an $n$-by-$n$ matrix with integer entries whose absolute values are bounded by a constant $C$. It is well-known that the entries of the inverse $A^{-1}$ can grow exponentially on $n$. (See ...
0
votes
0answers
55 views

Eigenvalues and eigenvectors of a very large sparse matrix

I have three questions about calculating the eigenvalues and eigenvectors of a very large sparse matrix. First of all, I have some large sparse matrices, some of which have only two nonzero diagonals ...
3
votes
1answer
61 views

Mapping inclusion theorem for the numerical range

We denote the numerical range of a complex square matrix $A \in \mathbb{C}^{n\times n}$ by $W(A)$. Let $A \in \mathbb{C}^{n\times n}$ and let $f: \mathbb{C} \to \mathbb{C}$ be, say, an entire ...
0
votes
0answers
41 views

The derivative in inverse matrix [on hold]

I wonder how to calculate the following derivative w.r.t to matrix: $\frac{d(x^TW^{-T}W^{-1}x)}{dW}$, where $W$ is a $\mathbb{R}^{d\times d}$ matrix and $x$ is a $\mathbb{R}^d$ vector. The result ...
1
vote
0answers
33 views

Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix

Lets assume we have the following equation: $AU=\lambda U \Rightarrow\left[ \begin{array}{c|c|c} 0 &A_{12}&A_{13}\\ \hline A_{21}& 0& A_{23}\\ \hline A_{31}&A_{32}&0 \end{...
9
votes
0answers
149 views

More mysterious properties of Gram matrix

This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question. The following fact could be extracted from 0402087: For any $a_i\...
6
votes
0answers
148 views

A curious $q$-identity

Let $[x]_{q}=\frac{1-q^x}{1-q}$ and $\binom{x}{n}_{q}$ denote a $q$-binomial coefficient. Let $A_n(x,q)$ be the $n\times n $ matrix with entries $$q^\binom{i-j}{2}\binom{i+j+x}{i-j+1}_{q},$$ $0 \le i,...
16
votes
1answer
235 views

Convex hull of all rank-$1$ $\{-1, 1\}$-matrices?

Consider the set $\mathbb{R}^{m \times n}$ of $m \times n$ matrices. I am particularly interested in properties of polytope $P$ defined as a convex hull of all $\{-1,1\}$ matrices of rank $1$, that is,...
1
vote
1answer
82 views

Matrix of powers of pairwise differences

Let $\underline{c}:=\left(c_1,\dots,c_n\right)$ be pairwise distinct complex numbers, and let $k$ be a non-negative integer. Define the matrix $A_{n,k}(\underline{c})$ to contain the $k$-th powers of ...
4
votes
1answer
136 views

Inverse of a matrix and the inverse of its diagonals

While researching a question, I faced with the following problem: I have to prove that for a positive definite matrix we have $${\mathbf n}^T {\mathbf R}^{-1}{\mathbf n}\geq {\mathbf n}^T {\mathbf D}...
2
votes
0answers
126 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 $P$-...
2
votes
1answer
78 views

Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
1
vote
1answer
103 views

Minimal value of matrix norm induced by a norm

Let $X$ be a finite dimensional Banach space and define a matrix norm $\| \cdot \|_{X}$ by $$ \| A \|_{X} = \sup_{x \ne 0} \frac{\|A x\|_{X}}{\|x\|_{X}} $$ where the matrix $A$ is interpreted as an ...
0
votes
0answers
19 views

Is the Weak Popov Form of a matrix over a polynomial ring unique?

The Weak Popov Form of a matrix is a type of normal form of the matrix. We can find some special properties by changing each matrix to this form by considering some elementary operations on the matrix....
1
vote
0answers
30 views

A recap of regularity of singular values as a function over M_n

So the core of the question is the study of the function $$ s :M_n(\mathbb R) \mapsto M_n(\mathbb R)$$ $$A \rightarrow s_n(A) $$ where $s_n(A)$ is the greatest singular value of A. I know there has ...
5
votes
1answer
278 views

Determining the primitive order of a binary matrix

Let ${\bf A}_n$ be an $2n \times 2n$ matrix that is defined as follows $$ {\bf A}_n=\left( \begin{array}{c} 0&0&\cdots&0&0&0&0&1&1\\ 0&0&\cdots&0&0&...
0
votes
0answers
24 views

Determinant of the Sum of a Left Circulant and a nonsingular matrix over field of order 2

For any matrix $A=(a_{ij})\in M_n(\mathbb{F}_2)$, denote $\bar{A}=(\bar{a_{ij}}),$ where $\bar{a}=0$ if $a=1$ and $\bar{a}=1$ if $a=0$. Suppose $n$ is a power of two. Let $A$ (left circulant) and $I'$ ...
0
votes
1answer
61 views

Number of symmetric matrix with fixed margins

I'm looking for the cardinality of the set of symmetric matrices with entries in $\mathbb{Z}^*$ (the nonnegative integers) and fixed margins $\mathbf{k}=(k_1,k_2,...,k_q)$. I've also seen them called ...
3
votes
1answer
57 views

Dimension of fixed vectors of a semi-linear operator

Let $L$ be a field with a field embedding $\sigma:L \rightarrow L$, and $K=L^{\sigma}$ be the fixed field of $\sigma$. For $A \in M_n(L)$ a matrix, consider the set $X=\{x \in L^n|Ax=\sigma(x) \}$ ...
8
votes
1answer
234 views

What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?

Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
5
votes
1answer
163 views

Perturbing a normal matrix

Let $N$ be a normal matrix. Now I consider a perturbation of the matrix by another matrix $A.$ The perturbed matrix shall be called $M=N+A.$ Now assume there is a normalized vector $u$ such that $\...
4
votes
2answers
80 views

Stable matrices and their spectra

I am a graduate student in engineering and we work a lot with so-called Hurwitz (or stable) matrices. A matrix in our terminology is called stable if the real part of the eigenvalues is strictly ...
3
votes
1answer
131 views

Real part of eigenvalues and Laplacian

I am working on imaging and I am a bit puzzled by the behaviour of this matrix: $$A:=\left( \begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 &...
3
votes
1answer
128 views

Spectrum of this block matrix

Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \...
4
votes
1answer
144 views

Bicommutant theorem for commutative operator algebras

Let $\mathcal{B}(H)$ denote the space of bounded linear operators on a complex Hilbert space $H$. The von Neumann bicommutant theorem says: Theorem. Suppose that $\mathcal{A}$ is a $C^*$-subalgebra ...
1
vote
0answers
48 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, ...
1
vote
0answers
111 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 $(1 + p^0 + p + \cdots + p^i)$, where $i \geq 0$. Suppose that further the $(m,n)$-component $a_{m,n}$ ...
0
votes
1answer
73 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 ...
1
vote
0answers
61 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
132 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
128 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 $A$ ...
5
votes
2answers
208 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
votes
1answer
355 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
130 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 ...
0
votes
0answers
53 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 $b_1\ge b_2\ldots \ge b_m>0;$ $...
1
vote
0answers
95 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 $b_1\ge b_2\ldots \ge b_m>0;$ ...
0
votes
0answers
37 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 (...
0
votes
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
331 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
37 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
votes
0answers
36 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 ...
0
votes
1answer
58 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 $B$ ...
1
vote
0answers
55 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}{\...
0
votes
0answers
26 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\}} := \{...
27
votes
1answer
600 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
308 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
187 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
197 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 ...