Questions tagged [matrix-theory]

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.

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6
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0answers
72 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):...
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1answer
66 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} ...
2
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1answer
77 views

Routh-Hurwitz criterion for matrices

The Routh-Hurwitz criterion explicitly specifies a finite set of inequalities on the coefficients of a polynomial, necessary and sufficient that all zeros lie in the unit circle or in the left half ...
3
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1answer
74 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 \...
2
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0answers
60 views

Uniform continuity of Cholesky decomposition

I am trying to figure out whether the Cholesky decomposition is uniformly continuous within a set of matrices with bounded entries. Specifically, I would like to derive a modulus of continuity for the ...
8
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1answer
184 views

Kulkarni-Nomizu square root of the Riemann tensor

Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
2
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138 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 ...
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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 ...
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1answer
112 views

Existence of matrices in the field $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$ and let $e_1$, $e_2$, $\ldots$, $e_{10}$ denote its rows. For $i\in \{1,5 \}$, ...
2
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1answer
66 views

Existence of a matrix in $\mathbb{F}_2$ with some invertibility properties

All the matrices in this statement are in the field $\mathbb{F}_2$. Let $I$ be the identity matrix of size $10 \times 10$. What are all the possible $n$ ($\geq 6$) for which there exists a matrix $X$ ...
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103 views

Upper bound on the sum of the smallest non-zero eigenvalues

Let $\mathcal A := \{ A_1, A_2, \dots, A_n \} \subset \Bbb R^{d \times d}$ be a set of symmetric and positive semidefinite matrices. For a matrix $A_k \in \mathcal A$, denote its (real) eigenvalues by ...
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34 views

Does there always exist a(n uniform) polynomial that makes a positive definite symmetric matrix with polynomial entries into a sum of squares?

Suppose that I have a square and positive definite for every evaluation $x\in\mathbb{R}^{n}$ symmetric matrix $M(x)\in(\mathbb{R}[x])^{s\times s}.$ Does there always exist a polynomial $p(x)\in\...
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102 views

Expressing a polynomial as the determinant of a matrix of linear forms

I have heard that it's a well known result (in theoretical computer science?) that if we have a polynomial $p(t_1,\dots,t_n)$ over $\mathbb Q$, we can find matrices $M_0,\dots,M_n/\mathbb Q$ such that ...
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37 views

For which sets of linearly independent complex matrices does there exist a vector whose image set under the given matrices is linearly independent?

For which sets $\{A_i \}_{i=1}^n \subseteq M_d$ of linearly independent $d\times d$ complex matrices does there exist a vector $v\in \mathbb{C}^d$ such that $$\text{dim} [\text{span}\{A_1v, A_2v, \...
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39 views

Spectral theorem for symmetric real tensors

Is there a definition of eigenvalues that allows to use a spectral theorem? Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\...
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51 views

Is it possible to reduce eigenvalues of tensors to an matrix eigenvalue problem?

Can we construct a larger matrix $M$ such that its eigenvalues are the same as the eigenvalues of a tensor $T$ of order 3? Let $\mathbf{T}$ be a fully symmetric tensor of order $3$ and size $N$. Its ...
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74 views

Distribution and expectation of inverse of a random Bernoulli matrix

This question cropped up as a part of my research. Let us assume a $n\times n$ random matrix $\mathbf{M}$ with elements iid distributed to a Bernoulli distribution that takes values $\{0,1\}$ with ...
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2answers
326 views

Rank of $A\otimes B - B\otimes A$

Let $A$, $B\in\mathbb{R}^{n\times n}$ be full-rank random matrices and define the Kronecker products $P=A\otimes B$ and $Q=B\otimes A$. Through example-based examination, I have found that $\text{rank}...
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1answer
716 views

How to see that the determinant of this matrix is nonzero for all primes?

I'm trying to show that $\sum_{i = 0}^{p-2} (i+1)^{-1} t^{i+n}$ where $0 \leq n \leq p-2$ spans the vector space $\mathbb{F}_p[t]/(1-t)^{p-1}$ as a rank $p-1$ module over $\mathbb{F}_p$. In other ...
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2answers
210 views

How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there?

How many non-isomorphic associative algebras of dimension 2 over the field F_{p^k} are there? As much as I have searched, I have not found any results that answer my question; not even for k = 1,2.
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64 views

Infinite positive matrices with probability eigenvector

Let $A$ be an infinite non-negative matrix with integer entries ($a_{ij} \geq 0, \forall i,j \in \mathbb N$). Suppose that $A$ is irreducible, aperiodic, and recurrent. So that it satisfies the ...
8
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1answer
244 views

General Sylvester's linear matrix equation

For what conditions on $A$, $B$ and $C$ (square matrices of size $n$) would there be a unique solution to $$ ABX + AXC + XBC = D, $$ for any $D$? Can one expect a characterization similar to the ...
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0answers
112 views

Which matrix decompositions feature permutation matrices?

It's well known that LU decomposition is only numerically stable if it's combined with row and/or column pivoting. It makes me wonder if there are other matrix decompositions that can profitably be ...
4
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1answer
92 views

How to compute an indefinite generalisation of QR decomposition

Given an arbitrary complex matrix $M$ and real, diagonal but possibly indefinite matrix $\Delta$, the problem is to solve the following system of equations: $$\begin{aligned} M^*\Delta M &= LD^2L^*...
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2answers
124 views

If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$

Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
3
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1answer
169 views

Does there exist an O(n^3) algorithm for deciding whether PAP^T = LDU is solvable given some square matrix A?

Let $A$ be an arbitrary real square matrix. Does there exist an $\mathcal O(n^3)$ algorithm for deciding whether there exists a permutation matrix $P$, lower unit triangular matrix $L$, upper unit ...
3
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0answers
96 views

Polynomiality of the inverse of equally weighted Varchenko matrices attached to hyperplane arrangements

Let $d\in \mathbb{Z}_{\ge 1}$, let $\sigma = (H_i)_{i\in \mathcal{I}}$ be a finite hyperplane arrangement in $\mathbb{R}^d$, where $H_i\subset \mathbb{R}^d$ is a hyperplane for $i\in \mathcal{I}$ (the ...
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1answer
94 views

For the purposes of solving linear equations, is there a fast decomposition that works for all Hermitian matrices?

Let $A$ be an arbitrary Hermitian matrix. Is there a way of efficiently factorizing $A$ for the purposes of solving $Ax = b$ for arbitrary $b$? There are two decompositions I'm aware of that nearly ...
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1answer
81 views

Are there any results in generalizing matrix theory to multidimensional arrays? [closed]

In matrix theory(2-dimensional arrays), we can define addition, multiplication, rank and determination etc. I'm working on generalizing these properties to multidimensional arrays as many as possible. ...
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39 views

Solving a Lyapunov inequality with a fixed block

Let $A\in\mathbb{R}^{n\times n}$ be the following partitioned matrix $$ A = \begin{bmatrix}A_{11} & A_{12} \\ A_{21} & A_{22}\end{bmatrix} $$ where $A_{11}\in\mathbb{R}^{n_1\times n_1}$, $A_{...
8
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1answer
116 views

Characteristic polynomial of a matrix related to pairs of elements generating $\mathbb{Z}/n\mathbb{Z}$

Fix $n\geq 2$. Let $A$ be the matrix whose rows and columns are indexed by pairs $(a,b)\in \mathbb{Z}/n\mathbb{Z}$ such that $a,b$ generate $\mathbb{Z}/n\mathbb{Z}$ (the number of such pairs is $\phi(...
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0answers
22 views

On the dimension of the projected rank $r$ matrix space

In $d$-dimensional complex number space $\mathbb{C}^{ d}$, we can define the rank at most $r$ matrix space $$ S=\{A|\ \mathrm{rank}(A)\leq r\}\subseteq \mathbb{C}^{d\times d}. $$ The dimension of ...
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0answers
41 views

Symmetric matrices of hyperbolic and elliptic type with certain kind of trace zero

I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question. Let $\...
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0answers
37 views

A lower-bound on matrix-function with vector product

I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
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0answers
37 views

Expectation of random matrix (minimum positive eigenvalue)

Assume, $M_i$ are random symmetric positive semi-definite matrices and there exists $ 0 < \mu_1 \leq \mu_2 $ such that $\mu_1 \|x\|^2 \leq x^TE[M_i]x \leq \mu_2 \|x\|^2$ holds for some $0 \neq x \...
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1answer
61 views

Determine unknown matrix function of particular form from known points

I encountered the following problem recently in a practical context. Fix $n \ge 1$. Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form $$ X \mapsto ...
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0answers
54 views

Projection onto column space perturbed by Gaussian noise

Suppose we have a matrix $X\in\mathbb{R}^{m\times n}$ (with $n \le m$) with iid standard Gaussian entries, and suppose we have noise matrix $W\in\mathbb{R}^{m\times n}$ with iid Gaussian entries, but ...
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0answers
73 views

How to find the similarity transformation of a matrix operator?

Consider the matrix operator $$ \mathcal{L}= \begin{pmatrix}{} -A\cdot \partial_x^2-(3\varphi^2-1)-2(\varphi',\partial_x\cdot)_{L^2}\varphi'' & c\partial_x\\ -c\partial_x & 1 \end{pmatrix} $$...
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3answers
192 views

Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$

Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such ...
7
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1answer
258 views

Closed form solution for $XAX^{T}=B$

Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$? $$X A X^{T} = B$$ Thank you.
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51 views

Derivative of matrix argument function with respect to eigenvalues of argument

Let $\mathsf{SPD}_n$ denote the set of all real symmetric and positive definite $n\times n$ matrices. This set is convex so for every $A,B\in\mathsf{SPD}_n$ there exists a smooth path $\varphi:[0,1]\...
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2answers
352 views

A question of invertibility of matrices

Let $A$ and $B$ be self-adjoint $n \times n$ matrices. Let $A$ be diagonal. Suppose $A+tB$ and $tA+B$ are invertible for all $t \in \mathbb R$. What can we say about $A$ and $B$? My guess is that $\...
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0answers
32 views

How to adjust two Gramian matrices to make them “aligned” without enumeration?

I have two set of vectors: V1,V2,V3,...,Vn W1,W2,W3,...,Wn And two Gramian matrices of them: Mv, Mw Now I want to find an ordering of W1,W2,W3,...,Wn such that the Gramian matrix of the new order of ...
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0answers
123 views

Schrodinger operator with matrix potential

This question is inspired by attempts to extend in some way Shen's 1999 "On fundamental solutions of generalized Schrödinger operators" to Schrödinger operators $- \Delta + V $ with some ...
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0answers
46 views

Find occurrences of certain matrix inside a matrix

This problem occurred from my need to find all graphs with a certain topology inside a bigger one. I don't need the subgraphs but the graphs that have the exact topology I am searching. We know for ...
2
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1answer
99 views

Bound for matrix inner product based on singular values

Regarding the matrix inner product based on singular values, Lewis (1995) "The convex analysis of unitarily invariant matrix functions" states the result by von Neumann that $\langle X,Y \rangle \leq \...
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0answers
108 views

Upper and lower bounds on the entries of a matrix power

Say I have a non-negative square $n\times n$ irreducible stochastic matrix $A$ (i.e., each column sums to 1), for which the following holds: $$A_{ij} > 0 \iff A_{ji} > 0.$$ I know that no more ...
3
votes
1answer
297 views

Inequality for $AB + BA$ when $A,B\geq0$, reference request

Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues. It is well-known that the eigenvalues of the expression $AB +...
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0answers
39 views

Minimum rank of a product of two block diagonal matrices with an arbitrary matrix

Let us assume that we have an arbitrary full-rank $l\cdot b \times l\cdot p$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), an $m \times ...
2
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0answers
85 views

Random sparse and invertible matrices

Let $n\leq m$ and $0\leq k\leq (n\times m - \min\{n,m\})$ be in $\mathbb{N}$. Let $\mu$ be a probability measure dominated by the Lebesgue measure on $\mathbb{R}$ and generate a random $n\times m$ ...

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