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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
110 views

Interpreting positive semidefinite matrix as a graph

Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
1 vote
0 answers
18 views

Correlation Matrix Problem of Three Decomposition Level of DWT

I'm trying to apply a DWT with 3 composition levels and the following question arose when calculating the composition matrix. The step I'm trying to follow is: The DWT coefficientes are obtained from ...
6 votes
2 answers
975 views

Matrix groups and presentation

Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group? More ...
2 votes
1 answer
2k views

power of a block triangular matrix

I have a matrix in the form : $$M = \begin{pmatrix} A & 0 & 0 \\\ B & A & 0 \\\ C & D & A \end{pmatrix} $$ where $A,B,C,D$ are diagonalizable square matrices and I want to ...
2 votes
2 answers
1k views

How to derive the mean of inverse-Wishart distribution?

How to derive the mean of inverse-Wishart distribution in Inverse-Wishart distribution? I have no idea about it. Thanks for your help.
5 votes
3 answers
228 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 ...
11 votes
2 answers
752 views

A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
1 vote
1 answer
116 views

Minimal number of linearly dependent rank-1 projectors

What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent. PS: the ...
2 votes
1 answer
1k views

How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?

$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists? I am interested in this question, because my ...
8 votes
2 answers
613 views

Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

It is known that an entire function that is nowhere zero must be the exponential of another entire function. Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...
0 votes
0 answers
118 views

On a matrix equation with Kronecker product

Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
8 votes
1 answer
311 views

Recovering eigenvalues of a matrix from its $p$th compound matrix

This question was motivated by Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?. Let $A$ be an $n\times n$ matrix over a field. Suppose we are given the $p$th ...
2 votes
1 answer
224 views

Continuous path of unitary matrices with prescribed first column?

Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$. Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
2 votes
1 answer
340 views

Simultaneous triangularization of two diagonal matrices and a symmetric matrix

I have the following quadratic eigenvalue problem: $\det(\lambda^2M + \lambda D + J)=0$ where, M and D are both $n \times n$ real diagonal matrix with positive diagonal entries; $J$ is 1) a ...
0 votes
1 answer
67 views

Inequality for extremal values of product of Hermitian matrices

I am looking for a reference to verify the following inequality, where $X$ and $Y$ are Hermitian positive semidefinite matrices: $$ \lambda_n(X^{1/2}YX^{1/2}) = \lambda_n(XY) \leq \lambda_n(X)\...
2 votes
0 answers
143 views

What are the name and inverse of an interesting integer matrix?

It is practicable to compute the matrix inverses \begin{align*} \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 2 & 2^2 \\ \end{pmatrix}^{-1} &=\begin{pmatrix} 1 & 0 &...
5 votes
2 answers
342 views

How expressive is $e^A$ in the sense of universal approximation?

For any real matrix $B\in\mathbb{R}^{n\times n},n\ge 2$ and precision $\varepsilon$, is there a real matrix $A\in\mathbb{R}^{n\times n}$ such that $\|e^A-B\|_F<\varepsilon$? ($F$ refers to ...
1 vote
0 answers
69 views

When can we separate two pairs in ${\mathbb H}_n$, although it is not a lattice?

Recall that a lattice is a partially ordered set $E$ for which any pair $a,b\in E$ admits a least upper bound and a greatest lower bound. Remark that given four elements $a_i,b_j$ ($j=1,2$), in order ...
2 votes
1 answer
111 views

Primal identity in matrix semigroup

Given a finite set of matrix $\{M_1,M_2,\cdots,M_n\}\subseteq \mathbb{C}^{d\times d}$, we consider the semigroup generated by matrix product. We call $s_1\cdots s_k$ an identity index if $M_{s_1}M_{...
0 votes
0 answers
19 views

Find condition of X such that I-XSA is nonsingular, where $S$ is skew-symmetric and $A$ is symmetric, nonsingular

Given $I_n$ is the identity matrix, $A \in \mathbb{R}^{n,n}$ is symmetric and nonsingular, and $B\in \mathbb{R}^{n,n}$ is skew-symmetric.\ a) For which condition of a matrix $X \in \mathbb{R}^{n,n}$, ...
2 votes
0 answers
119 views

Decompose a rational matrix as an integer matrix and an inverse of integer matrix

Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
1 vote
2 answers
115 views

Methods to solve for a matrix whose entries satisfy certain properties

(This question is a repost of a deleted question I asked, because the previous version had several elements missing) Setting For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
14 votes
1 answer
624 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):...
1 vote
0 answers
119 views

Some kind of product of two 2d tensors to create a 3d tensor?

I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays): given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
9 votes
2 answers
316 views

Almgren's regularity Theorem ; a simple example?

Let me remind Almgren's regularity Theorem: the singular set of area-minimizing surface has codimension at least $2$. I wish to share here a simple example in low dimension, although I don't know ...
0 votes
0 answers
85 views

Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix

Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is $$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\ ...
8 votes
2 answers
506 views

Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices

My question is motivated by this one, but within real matrices instead of complex ones. ${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
0 votes
0 answers
44 views

A particular selection of rows in upper triangular matrices

Let $A$ be a strictly upper triangular $n\times n$ matrix whose entries are either 0 or 1 (diagonal entries are all 0) with the nullity $m<n$. Let us denote $R_j$ and $C_j$ with the rows and ...
2 votes
1 answer
156 views

Matrix inequalities in series form

While studying the positivity of mechanical systems, we land on the following conjecture but don't know how to prove this. If a square matrix $A \in \mathbb{R}^{m\times m}$ satisfies both the two ...
4 votes
1 answer
438 views

A potential new norm for matrices and Horn's inequalities

I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
3 votes
2 answers
233 views

Generating $\mathbf{PGL}_2(\mathbb{Z})$ and $\mathbf{PGL}_2(\mathbb{Q})$

It is well known that $\mathbf{PGL}_2(\mathbb{Z})$ is finitely generated, and that $\mathbf{PGL}_2(\mathbb{Q})$ isn't. My question is: what is a fast, natural way to see these properties without ...
2 votes
4 answers
283 views

Find a square, stochastic matrix of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix $M$ couldn't be primitive, so there would be at least one entry equal to zero in every power $M^k$ (Perron-Frobenius theory). Preferably the ...
0 votes
0 answers
50 views

Rank of Coxeter matrix

Let $Q$ be a quiver and $\Phi_Q$ be the Coxeter matrix of $Q$. Then $\Phi_Q\pm I$ are full-rank?
4 votes
0 answers
95 views

A question on products of linear combinations of complex matrices

Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds $$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
3 votes
2 answers
3k views

Inverse of particular lower triangular matrix

I have an $n \times n$ lower triangular matrix $A$ where $$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$ $$A_{ii}=1, \quad 1 \leq i \leq n,$$ and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
0 votes
1 answer
184 views

Faulty algorithm for simultaneous diagonalization?

I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
2 votes
0 answers
126 views

Visualisation of general 3x3 matrices, with applications to the pedagogy of linear algebra?

I've got a method for visualising non-zero $2 \times 2$ real matrices (modulo non-zero scalar factor) using the fact that: Nonnegative determinant matrices (modulo non-zero scalar factor) are in 1-to-...
1 vote
1 answer
204 views

Third order matrix differential norm

Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
3 votes
0 answers
138 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
0 votes
0 answers
100 views

On the exponentiation of a stochastic matrix where the exponent is a function of matrix size

In this question, I asked about any arbitrary stochastic matrix $A(n)$ of the particular form $$A(n) = \begin{pmatrix} 1 & 0 & \cdots & 0\\ x_{21} & x_{22} & \cdots & 0\\ \...
1 vote
0 answers
79 views

Dynamical formulation of the 2-Wasserstein distance for *discrete* matrix-valued measures

TL;DR: I want to find a definition generalizing "$t \mapsto \frac{1}{m} \sum_{k = 1}^{m} \delta_{x_k(t)}$ is a Wasserstein gradient flow" to matrix-valued probability measures. Let $(X, d)$ ...
3 votes
1 answer
128 views

On the bounds of the sum of the squares of spectral variation of two real symmetric matrices

Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
8 votes
7 answers
1k views

One observation of special type of square matrix exponentiation

I was studying the following type of matrices, $$ A = \begin{pmatrix} 1 & x_{12} & \cdots &x_{1n}\\ 0 & x_{22} & \cdots &x_{2n}\\ \vdots\\ 0&\cdots&0&x_{nn} \end{...
3 votes
1 answer
179 views

The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$

Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
3 votes
1 answer
105 views

Matrix inequality in a paper by Piccinini-Spagnolo

In the paper 'On the Holder continuity of solutions of second order elliptic equations in two variables' by Piccinini and Spagnolo, they prove the following estimate: $$ \begin{array}{ll} \left(\int_S ...
1 vote
0 answers
61 views

Solve linear matrix equation involving convolution

I am facing following equation: $$ A * X + C \cdot X = D $$ with: $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure, $X \in \mathbb{R}^{n \times n}$ the ...
2 votes
2 answers
819 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 ...
4 votes
1 answer
265 views

Positive system of algebraic integers

Let $\mathbb{A}$ be the ring of algebraic integers. Consider a sequence $(d_i)_{i \in I}$, with $I$ a finite set and $d_i \in \mathbb{A} \cap \mathbb{R}_{\ge 1}$, such that $$d_i d_j = \sum_{k \in I} ...
1 vote
1 answer
130 views

eigenvalues of matrices (with positive entries)

I am reading an old paper by Kawpien and Pelczynski, Studia Math. 1970. It claims that singular values of a matrix (with positive entries? I am not sure) is given by $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
1 vote
1 answer
69 views

A question about the sign of quadratic forms on nonnegative vectors

Let $M$ be a real square matrix of order $n\ge 3$. Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$. Can ...

1
2 3 4 5
10