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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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 ...
Dragnovith's user avatar
1 vote
0 answers
86 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 ...
wsz_fantasy's user avatar
8 votes
2 answers
610 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 ...
Kanghun Kim's user avatar
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, ...
mukhujje's user avatar
  • 281
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 ...
ccriscitiello's user avatar
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)\...
turtlesandwich's user avatar
2 votes
0 answers
141 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 &...
qifeng618's user avatar
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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 ...
li ang Duan's user avatar
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0 answers
18 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 $X \in \mathbb{R}^{n,n}$, ...
IscoBerlin's user avatar
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116 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}(\...
ghc1997's user avatar
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1 vote
2 answers
113 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 $...
algebroo's user avatar
  • 113
1 vote
0 answers
118 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 $B_{n\times p}$, ...
Min Wu's user avatar
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1 answer
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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 ...
Alm's user avatar
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0 votes
0 answers
83 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 \\ ...
KAJ226's user avatar
  • 131
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 $\...
Denis Serre's user avatar
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42 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 ...
ABB's user avatar
  • 3,898
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 ...
Denis Serre's user avatar
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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 ...
IscoBerlin's user avatar
9 votes
2 answers
315 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 ...
Denis Serre's user avatar
  • 51.5k
4 votes
1 answer
437 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 ...
Pedro Poitevin's user avatar
3 votes
2 answers
232 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 ...
THC's user avatar
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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?
user145752's user avatar
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)^{...
user493645's user avatar
0 votes
1 answer
182 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 ...
TobiR's user avatar
  • 103
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-...
wlad's user avatar
  • 4,792
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\\ \...
Subhankar Ghosal's user avatar
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 ...
André Schlichting's user avatar
1 vote
0 answers
78 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 $(X, d)$ ...
ViktorStein's user avatar
2 votes
1 answer
110 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_{...
gondolf's user avatar
  • 1,483
1 vote
1 answer
198 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 ...
RS-Coop's user avatar
  • 39
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 \...
shahulhameed's user avatar
8 votes
7 answers
999 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{...
Subhankar Ghosal's user avatar
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?
ABB's user avatar
  • 3,898
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 ...
Adi's user avatar
  • 483
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 ...
JannyBunny's user avatar
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 $t_i=\sqrt{\sum_{j\ge 1}a(i,j)^2}$....
user92646's user avatar
  • 617
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} ...
Sebastien Palcoux's user avatar
1 vote
1 answer
68 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 ...
user139975's user avatar
2 votes
0 answers
42 views

Robustness of largest singular vectors with respect to noise

I would like to find a result that shows that the largest right-singular vectors of a data matrix are in some sense robust with respect to low-variance noise perturbations. Specifically, let $X = U D ...
foobar_98's user avatar
1 vote
0 answers
118 views

Convex matrix combination

We all know the notion of a convex combination like $$\lambda x_1 \; + \; (1 - \lambda) x_2$$ for some $\lambda \in (0, 1)$. However, I am trying to find literature where this concept has been ...
foobar_98's user avatar
0 votes
1 answer
242 views

Are there zero entries in the eigenvector corresponding to a simple eigenvalue?

For a real symmetric matrix $M$ and a simple eigenvalue $\lambda$, under which conditions the corresponding eigenvector has no zero entries? Perhaps, this is unconditional and one can provide a proof?
Vladimir's user avatar
1 vote
0 answers
84 views

Solving a block tridiagonal system with diagonal perturbations

Say we have a block tridiagonal matrix, $T \in \mathbb{R}^{NL \times NL}$, with constant off diagonals, $\mathbf{B} \in \mathbb{R}^{L\times L}$, given by $$ T = \begin{bmatrix} \mathbf{A}_1 & \...
matthewd49's user avatar
9 votes
1 answer
242 views

Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces

$\DeclareMathOperator\GL{GL}$We will call a subset $S \subset \GL_n(\mathbb{R})$ a linear subspace if it is of the form $S = S'\cap \GL_n(\mathbb{R})$ for some $S'\subset M_n(\mathbb{R})$ which is a ...
user49822's user avatar
  • 2,033
1 vote
0 answers
183 views

Schatten norm inequality

Let $A,B$ be two $n\times n$ matrices. Find a lower bound of the $p$-th Schatten norm $\|(A-B)(A-B)^\ast\|_{S_{p/2}}^{1/2}$ in terms of Schatten norm of $\|(AA^*+BB^*)\|_{S_q}$ for any relation ...
volond's user avatar
  • 97
8 votes
0 answers
405 views

Semigroups of matrices closed under conjugate transposition

An involution semigroup or $\star$-semigroup is a unary semigroup $\langle S,{\cdot}\,,{}^\star\rangle$ that satisfies the equations $$ (x^\star)^\star = x \quad \text{and} \quad (xy)^\star = y^\star ...
E W H Lee's user avatar
  • 563
0 votes
0 answers
141 views

Multiplying by Loewner-ordered matrices

Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
Athere's user avatar
  • 89
2 votes
1 answer
262 views

Classification of congruent integer matrices

I am interested in the following question: Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...
WhenYouHaveNoClue's user avatar
1 vote
1 answer
128 views

The number of invertible 4×4 circulant matrices over the ring Z

Is the number of invertible 4×4 circulant matrices over the ring of integers Z finite? I am looking for a reference which discusses this case.
Katy's user avatar
  • 11
1 vote
0 answers
217 views

Majorization for singular values of the difference of two matrices: $|\sigma(A)-\sigma(B)| \prec_w \sigma(A-B)$?

For two vectors $x$ and $y$ in $\mathbb{R}^n$, recall that $y$ weakly majorizes $x$, denoted by $x\prec_w y$, if the sum of the $k$ largest entries of $x$ is smaller than or equal to that of $y$ for ...
Nuno's user avatar
  • 213
1 vote
0 answers
40 views

What is the complexity of the matrix multiplication closure for a given generating system?

Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
ArminJR's user avatar
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