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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0answers
35 views

Matrix irreducibility [closed]

Consider irreducible nonnegative matrix $\mathbf{A} \in \mathbf{M}_{n}(\mathbb{R})$ such that $a_{ij} \in [0,1)$ as element of $\mathbf{A}$ of period $p$. If $\mathbf{A}^{T}$ is transpose of $\mathbf{...
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43 views

For a closed, bounded set of matrices, are the corresponding entropies also bounded?

Suppose we have a closed, bounded subset $S \subset \text{Mat}_{n\times n}(\mathbb{R}_{> 0})$, and we define the entropy of a matrix $M \in S$ as $$H(M) = \sum_{i,j} M_{i,j}\log\left( \frac{1}{M_{i,...
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121 views

Reference for $2$-by-$2$ integer matrices

Currently I am studying number theory, in particular modular functions and imaginary quadratic fields. I am looking for a reference on $2$-by-$2$ integer (or rational) matrices, especially: ...
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55 views

System of quadratic equations

Let $B_1,\ldots,B_s$ be $(s\times s)$ symmetric real matrices and $x=\left(x_1,\ldots,x_s\right)^\prime$ a $(s\times 1)$ vector of unknowns. Is there a way or reference theory for studying the ...
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52 views

Eigenvalue perturbation Problem

Consider a nonnegative matrix $\mathbb{K} \in \mathbf{M}_{n}(\mathbb{R}) $ with positive diagonal entries, which is perturbed by a small nonnegative matrix $\mathbb{E} \in \mathbf{M}_{n}(\mathbb{R}) $ ...
2
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1answer
64 views

Average the covariance matrix over all orthogonal matrices

Let $M=O\Lambda O^\top$ be a positive semi-definite matrix, where $\Lambda\in \mathbb{R}^{p\times p}$ is a diagonal matrix with non-negative entries and $O\in \mathbb{R}^{p\times p}$ is an orthogonal ...
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0answers
63 views

Linearly independent vectors from a matrix product

I have a product of $n$ integer-valued matrices: $$V=M_1 M_2 \dotsm M_n \,.$$ The $M_i$ are not square matrices, but the rows of $M_i$ and the columns of $M_{i+1}$ have the same length, so the ...
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2answers
179 views

Is it possible for the reduction modulo $p$ of an non-commutative semisimple algebra to be commutative?

Suppose that $I, X_1, \ldots, X_{d-1}$ are $n \times n$ matrices with integer entries whose $\mathbb{Z}$-span is a subalgebra of $\mathrm{Mat}_n(\mathbb{Z})$. Suppose that, thought of as a subalgebra ...
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57 views

Complexity of a combination of matrix operations [closed]

Suppose we have the following matrices: $$\vec{K}\text{: an $N$-by-$N$ matrix}$$ $$\vec{I}\text{: the $N$-by-$N$ identity matrix}$$ $n$, $N$ are constant numbers. I want to estimate the complexity of ...
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61 views

Distance function on generalized upper half planes

Let $\mathbb{H}^n$ be the quotient $GL_n(\mathbb{R})/O(n) \cdot \mathbb{R}^\times$, which at least in the theory of automorphic forms is called a generalized upper half plane. We could also think of $...
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12 views

Singular values of a matrix that its rows have (1) a tightly bounded angle, and (2) at least some norm

We're looking for a connection between a matrix's singular values and some information we hold about its rows. We wish to find a tight bound on the singular values. We know on the rows that they have (...
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74 views

Is this “semi-tensor product” something recently invented? Are there other usages of it?

The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
8
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1answer
149 views

On the coefficients that appear in finite groups of matrices with integer entries

Let $n$ be a positive integer and $G$ be a finite group of $n\times n$ matrices with integer coefficients, i.e. $G\subset\operatorname{GL}_n(\mathbb{Z})$. It is known that for sufficiently large $n$, ...
4
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56 views

Minimal set generators ideal submaximal minors

Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as: $$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...
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1answer
73 views

Eigenvalues of a block matrix with zero diagonal blocks

Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix \begin{equation} M:=\begin{pmatrix} 0_{k_1} & A\\ A^\top & 0_{k_2} \end{pmatrix}, \end{equation} ...
10
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2answers
430 views

Are the trace relations among matrices generated by cyclic permutations?

Let $X_1,\dots,X_n$ be non commutative variables such that $\operatorname{tr} f(X_1,\dots,X_n) = 0$ whenever the $X_i$ are specialized to square matrices in $M_r(k)$ for any $r \geq 1$. Does this ...
4
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1answer
124 views

Can all finite-dimensional noncommutative algebras with trace be trace-preserving embedded into matrix rings?

Suppose I have a finite (non-)commutative ring $R/k$ (over a field $k$ of char $0$) with a linear "trace" function $t: R \to k$. Can I always find an embedding $f: R \to M_r(k)$ compatible ...
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35 views

maximize quadratic form with respect to cov matrix of multinomial and linear constraint

Given $x_{m,1}, T$, how to solve $\max_{y} x^T(\mathop{\mathrm{diag}}(p)-pp^T)x$, s.t. $p = Ty, \textbf{1}^Ty=1$ The dimensions are $p_{m,1}, y_{n,1}, T_{m,n}$ and $T$ is a transition matrix ...
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18 views

Approximating singular values of the resolvent matrix for a non-Hermitian matrix

I have a pretty niche question that stems from the following answer: https://mathoverflow.net/a/79129/87974 I am interested in bounding the following quantity: $b := |e_k^T R(z)^T R(z) e_k|$, where $...
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0answers
51 views

Inverting “codimension matrix” for polytopes?

Let $P$ be an abstract polytope. Let's construct its square matrix $A$ as follows. Its lines and columns are labelled by all faces of $P$, of all dimensions. Put $A(F_1,F_2)=t^m$ if $F_1$ is a subface ...
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1answer
90 views

The rank of the Hadamard product

For matrices $D\in C^{d×p}$ and $E\in C^{d×p}$ with $d>p$, if $D$ is a full column matrix, for what condition that $D \odot E$ is also a full column matrix where $\odot$ denotes the Hadamard ...
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1answer
52 views

What is the spectral norm of the matrix $\text{diag}(p)-pp^T$?

I have a probability vector $p$ s.t. $1^Tp=1$ and $p\geq 0$. I want to compute the spectral radius of the matrix $M=\text{diag}(p)-pp^T$ where $\text{diag}(p)$ has its diagonal elements as $p$ and its ...
0
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1answer
122 views

Localizing the intersections of cubics

For Hermitian matrices $A,B \in \mathbb{C}^{n \times n}$, can one readily compute a set of cones that separate the maxima of $$\frac{x'Ax}{x'Bx}$$ among $x$ with unit-norm components? i.e. where do ...
3
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1answer
76 views

Operator norm of difference of matrix decompositions

This question is in part related to a question that I have already posed. Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
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0answers
24 views

Inverse rotation of Axis/Angle [closed]

I have 3d objects which are translated at certain z from each other and rotated. The information of the rotations are saved in the vector as AxisAngle format: {x_vec, y_vec, z_vec, teta_angle}. So I ...
2
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1answer
142 views

The “best way” to order unknowns in linear systems

Start with a linear system of the form \begin{equation*} Ax + Bt + C = 0, \end{equation*} where $x = (x_1, \dots, x_n) \in \mathbb R^n$ is the vector of unknowns, $t \in \mathbb R^m$ is a vector of ...
10
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2answers
353 views

A specific coset decomposition of $\mathrm{GL}_n(\mathbb{C})$

Disclaimer: I am a theoretical chemist (not a mathematician). I have tried asking this question at Math SE with no luck (https://math.stackexchange.com/questions/4080696/a-specific-coset-decomposition-...
2
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1answer
194 views

Operator norm of triangular truncation on symmetric matrices

Inspired by this question. It is known that for the matrix $T_n \in \mathcal{M}_n$ (the space of real-valued $n \times n$ matrices) defined by \begin{equation*} (T_n)_{ij} = \begin{cases} 1 & i \...
2
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0answers
141 views

System of matrix equations

Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$ Question: Is ...
4
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2answers
318 views

Sufficient conditions for invertibility of a block tridiagonal matrix

Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal matrix: $$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...
-2
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1answer
144 views

Derivative of log determinant [closed]

Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative? $$ \frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right). $$
4
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2answers
85 views

Hankel determinant of incomplete gamma functions

I have some expressions that involve Hankel determinants of incomplete gamma functions. They are of the ($r \times r$ form) I'd like to evaluate these determinants. Elementary operations help, but ...
1
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1answer
46 views

Upper bound on the size of vectors contained in an ellipsoid?

Crossposted at Math SE Consider the diagonal matrix $$D=\left[\begin{array}{cccc} 1^{- 2 p} & 0 & \cdots & 0 \\ 0 & 2^{-2 p} & \cdots & 0 \\ \vdots &...
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0answers
53 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 ...
3
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1answer
85 views

Subalgebras of the Temperley-Lieb algebra

I've recently met with the Temperley-Lieb algebra in my work. I'm in no way a specialist, and it's seems like a pretty simple question, but nevertheless. I'm interested in the subalgebra generated by ...
1
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1answer
94 views

Is trace of a slice of an elementary function of a matrix also elementary?

Let we have an elementary function $f(W)$, applicable to a matrix. Now consider the function $g(x)=\operatorname{tr} f(W+x),$ where $x$ is scalar. Is $g(x)$ necessarily an elementary function? Simple ...
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0answers
28 views

Is there an efficient way to do semidefinite programming with a Lyapunov equation constraint?

I am trying to numerically solve semidefinite programs of the form $$\begin{array}{ll} \underset{X,Y}{\text{minimize}} & \operatorname{tr}(AX)\\ \text{subject to} & BY + YB = X\\ & X, Y \...
2
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2answers
126 views

growth of the permanent of some band matrix

Consider such special band matrix of dimension $n$. It is a $0-1$ matrix, and only the first few diagonals are nonzero. Specifically, $$ H_{ij} = 1 $$ if and only if $|i-j| \leq 2$. How does the ...
1
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1answer
92 views

Matrix equation involving quadratic form

Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let $$A_i(\alpha) = (X\Lambda(\alpha) X^...
2
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1answer
70 views

Judge a special positive definite matrix in probability

Assume $\mathbf{x}$ is a random vector. The question is to judge whether $$E \{ (\mathbf{xx'})^{-1} \}- E\{(\mathbf{xx'})\}^{-1}$$ is positive definite or not. I have no idea how to do it. Could ...
3
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1answer
59 views

The covariance matrix of quadratic form, without normal assumption

Assume $\mathbf{x}$ is a random vector with mean $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. Symmetric matrices $\mathbf{A}$ and $\mathbf{B}$ are given. Without assuming normality, how to ...
1
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0answers
34 views

Growth of the number of columns $j=1,\dotsc,p$ such that $\|Ae_j\|_1 > p^\alpha$ for symmetric $A$ with bounded spectrum?

Consider the set $\mathcal S(p)$ of symmetric matrices $A$ of size $p\times p$ with bounded spectrum, say, $\lVert A\rVert_\text{op}\le 10$ and $\lVert A^{-1}\rVert_\text{op}\le 10$. Let $\alpha>0$ ...
3
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0answers
31 views

Stability of matrix equation

Let $M=I+A\in \mathbb{R}^{n\times n}$ for a skew-symmetric matrix $A$ with $\|A\|<1$ in the spectral norm. Using the $LU$-decomposition of $M$, it is easy to construct a solution $L,U\in \mathbb{R}^...
9
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2answers
336 views

Lower eigenvectors of nonnegative matrices with zero trace

Let $A$ be an $N\times N$ nonnegative matrix with all diagonal entries equal to zero and such that there is $n_0$ such that all entries of $A^{n_0}$ are strictly positive. Let $\lambda_1,\ldots, \...
5
votes
1answer
214 views

Do there exist positive definite matrices $A$ and $B$ satisfying this condition?

Denote by $\mathbb{P}_n$ be the set of real symmetric positive definite $n \times n$ matrices. In $\mathbb{P}_n$, we define an partial order as follow: for $X, Y \in \mathbb{P}_n$, we say $X \prec Y$ ...
3
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0answers
91 views

Multiplication of two Pauli string

Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $ Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $. Here $I,X,Y,Z$ are Pauli matrices defined explicitly as: $$ I = \begin{...
4
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1answer
116 views

On the real and finite field rank of a $0/1$ matrix - I

Let $M\in\{-1,0,+1\}^{n\times n}$ be a matrix of rank $r$. Consider the matrix $f(M)\in\{0,+1\}^{mn\times mn}$ where $0$ in $M$ is replaced by $m\times m$ all $0$ matrix, $+1$ in $M$ is replaced by $m\...
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1answer
128 views

If the direct sum of $L$ and $M$ has a pseudoinverse, then do $L$ and $M$ have pseudoinverses?

Let $L$ and $M$ be matrices over a commutative ring $R$ equipped with an involution "$*$". Define $L \oplus M$ (the "direct sum" of $L$ and $M$) to be $\begin{bmatrix}L & 0 \\ ...
2
votes
1answer
136 views

Block circulant matrix with some non-zero minors

Consider the following block circulant matrix over the field $\mathbb{F}_2$ \begin{equation*}M:= \begin{pmatrix} B_0 & B_1 & B_2 & B_3 & B_4 \\ B_4 & B_0 & B_1 ...
5
votes
2answers
99 views

Distance of low-rank matrices to the identity for the $\infty$-norm

I am trying to get a lower bound (or even the exact value) of $$ \min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m $$ where $m \leq n$, and the ...

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