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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-5 votes
0 answers
32 views

Chain Rule of Quadratic Matrix? [closed]

Hi I want to find an easy way to get the derivative of this function expressed in matrix format. What is results of df/dy? Thanks a ton!! χ and y are vectors Λ and Ω are square matrices f(y)=(χ-Ωy)'Λ(...
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 ...
3 votes
1 answer
105 views

The rank of a certain linear combination of mutually commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be mutually commuting $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers. For any complex number $c$, let $A(c):=A_0+cA_1+c^2A_2+\ldots +c^rA_r$. We ...
-1 votes
0 answers
34 views

simpler question from sylvesters eqn [closed]

i think i can simplify my problem from sylvesters equation (AX - XB = C) to this: (D_1) X - X(D_2) = 0 where D_1 is diagonal nxn D_2 is diagonal mxm X is nxm D and D_2 are diagonal matrices and they ...
2 votes
0 answers
44 views

Relationship between the homology of two types of tensor products of $\mathbb{Z}/ 2 \mathbb{Z}$-graded objects?

Let's consider a $2$-periodic complex $F$ of free $R$-modules, which is just a $\mathbb{Z} / 2 \mathbb{Z}$-graded complex $$F_1 \xrightarrow{d_1} F_0 \xrightarrow{d_0} F_1$$ (really the arrow $d_0$ ...
1 vote
1 answer
33 views

Iteration matrix representation with complex conjugate operator

I am studying the convergence of a particular class of radial power flows, whose goal is to obtain the voltage solution for a given electric grid, i.e., a complex vector $\mathbf{V}$ that gives the ...
0 votes
1 answer
72 views

Every square complex matrix is similar to a TRIDIAGONAL complex-symmetric matrix?

Every square complex matrix is similar to a complex-symmetric matrix. But I think a stronger statement is also true: Every such matrix is also similar to a TRIDIAGONAL complex-symmetric matrix. Where ...
0 votes
1 answer
146 views

“Smallest” non-zero linear combination of vectors to obtain a non-negative vector

We say that a vector $\mathbf{x}$ in $\mathbb{Z}^j$ is non-negative if it is of the form \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_j \\ \end{bmatrix} where $x_{j} \geq 0$. Suppose that ...
2 votes
1 answer
250 views

Continuity of eigenvector of zero eigenvalue

Wonder whether anyone has an idea on showing the following or to point out that it is not true: Let $A(t) \in \Re^{n \times n}$ be differentiable over an interval $I$, and it has a zero eigenvalue for ...
3 votes
2 answers
376 views

Monotonicity of matrix conjugation

Let $A$ and $B$ be positive-definite matrices such that $A \le B.$ By matrix monotonicity of the root, this also implies that $A^{\alpha} \le B^{\alpha}$ for $\alpha \in [0,1].$ I am now curious under ...
1 vote
2 answers
112 views

Property for bounding matrix exponential

Wikipedia states in the exponential map section about the exponential of a matrix that for any matrices $X$, $Y$ it holds that $\|e^{X+Y}-e^{X}\| \leq \|Y\|e^{\|X\|} e^{\|Y\|}$ where $\|\cdot\|$ ...
-1 votes
0 answers
77 views

How to find two sets of vectors which satisfy a set of matrix equations [migrated]

In my trial to solve a system of matrix equations, I wish to find two sets of non-zero vectors of $\mathbb{R}^3$ (which may be not unique) $\{ A_i \}$ and $\{ B_i \}$ where $i \in I$ (an index set, ...
3 votes
0 answers
47 views

About a circular variant of Vandermonde matrix

Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
4 votes
2 answers
121 views

What does the matrix-valued solution of $X = A X A^T + \operatorname{Id}$ look like?

Let $A \in \mathbb{R}^{n \times n}$ be an invertible contraction, i.e. all singular values are in $(0,1)$. By reformulating the equation \begin{align*} & X = A X A^T + \operatorname{Id} \tag{1} \...
6 votes
1 answer
373 views

Arithmetic properties of positively reduced $2\times 2$-matrices

Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$ positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces at ...
2 votes
1 answer
294 views

$\omega$-Commuting matrices vs Stone-von Neumann Theorem

Let me first recall the Stone-von Neumann theorem that if two one-parameter groups of unitary operators $U_t$ and $V_s$ over a Hilbert space satisfy $U_tV_s=e^{ist}V_sU_t$ for every $s,t\in{\mathbb R}$...
126 votes
13 answers
26k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
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 ...
5 votes
2 answers
337 views

Existence of a specific stochastic matrix

Let $0\le x_1\le x_2\le \cdots\le x_n\le n-1$ be given. My question is as follows : Under which condition there exists a doubly stochastic matrix $M=(m_{i,j})_{1\le i,j\le n}$ s.t. $$\sum_{j=1}^n (j-1)...
0 votes
0 answers
34 views

some problem about the discrete of the first derivative operator

I am reading a paper (Parameter Choice Strategies for Multipenalty Regularization Massimo Fornasier, Valeriya Naumova, and Sergei V. Pereverzyev SIAM Journal on Numerical Analysis 2014 52:4, 1770-1794)...
0 votes
2 answers
57 views

Nondegeneracy of dominant singular value and positivity of dominant singular vector of connected nonnegative matrix

Call a (not necessarily square) nonnegative matrix $M$ connected if there do not exist permutation matrices $P$ and $Q$ such that $PMQ=\begin{pmatrix}A&0\\0&B\end{pmatrix}$ for some $A$ and $B$...
3 votes
0 answers
73 views

A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal

I apologize if this is too elementary a question, but I have not been able to make much progress. Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
0 votes
1 answer
61 views

Handling the $\ell^2$ norm of a matrix expression in a linear regression

I am reading a scientific article in which matrices are handled (which I do not use often). We consider a matrix $X\in\mathbb R^{n\times p}$ and a vector $y\in\mathbb R^n$. The authors show that the ...
0 votes
0 answers
42 views

Find condition on signed matrix $A$ such that $\sum_jA_{ij}\cos(\theta_i-\theta_j)\geq 0$ implies $\sum_j\cos(\theta_i-\theta_j)\geq 0$?

As stated in the title, My question is: Suppose $\sum_jA_{ij}\sin(\theta_i-\theta_j)=0,\forall i\in[n]$. How to find condition on symmetric, diagonal-free, signed matrix $A$ ($A_{ij}=+1$ or $-1$ if $i\...
6 votes
0 answers
187 views

Zero-one pairings between sets of vectors

Let $A\subseteq V$ and $B\subseteq V^\star$ be spanning sets in a finite-dimensional real vector space $V$ and its dual $V^\star$. Suppose that $$ \langle b,a\rangle\in\lbrace0,1\rbrace $$ for all $a\...
2 votes
1 answer
141 views

Upper bound for the rank of a Gram-type matrix

Let $V = (v_{1},...,v_{N})$ and $W = (w_{1},...,w_{N})$ be 2 sets, each containing $N$ vectors from $\mathbb{R}^{n}$; i.e, $v_{j}, w_{j} \in \mathbb{R}^{n}$ for all $1 \leq j \leq N$. Assume that $N$ ...
1 vote
2 answers
144 views

Right inverse of integer matrix

If I have a rectangular matrix $A$ (say $4 \times 6$) with integer entries, is there a way to tell whether it has a right inverse that also has integer entries. I know that if $AA^T$ has determinant $...
10 votes
3 answers
2k views

Trace inequality for non-reversible Markov chain

Let $P \in \mathbb{R}^{d \times d}$ be the transition kernel for a Markov chain with stationary measure $\pi$ and define $P^\ast$ to be the time-reversed transition kernel defined by $P^\ast_{ij} := ...
-2 votes
1 answer
130 views

Norm of a matrix function of a vector $x$ [closed]

I have a matrix $$A(x) = \frac{-1}{(1+\|x\|_2^2)^{\frac{3}{2}}}xx^T + \frac{1}{(1+\|x\|_2^2)^{\frac{1}{2}}}I$$ I see that $$\|A(x)\|_2 \le 1 \ \forall x$$ But is $\|A(x)\| \le 1$ in general $\forall x$...
-2 votes
3 answers
2k views

Convergence of a markov matrix

Consider a markov chain matrix P of size n x n (n states). P is known to be: 1- Not irreducible (i.e. there exist at least a pair of states i, j such that we cannot go from i to j) 2- Not all ...
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 ...
3 votes
1 answer
222 views

Cofactor matrices and positive semi-definiteness

If $A$ is an $n\times n$ matrix with real entries, let me write $\widehat A$ its cofactor matrix. Since the map $A\mapsto\widehat A$ is polynomial, homogeneous of degree $n-1$, it can be multi-...
0 votes
0 answers
68 views

Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
5 votes
1 answer
212 views

Are there any applications of the algebraic polar decomposition?

One of the decompositions mentioned in the Wikipedia page on matrix decompositions is the algebraic polar decomposition. This factors a square complex matrix $M$ into $M = SQ$ where $S = S^T$ and $QQ^...
6 votes
3 answers
573 views

Reference for partial Hadamard matrices

Definition. An $m\times n$ matrix is said to be a partial Hadamard matrix (let's say PHM) if its entries are chosen from $\lbrace -1, 1 \rbrace$ such that the dot product of each pair of row vectors ...
1 vote
1 answer
165 views

Matrices over a finite field: matrices for which some unipotent $U$ satisfies Trace$(ZU)=0$ for all $Z$ in the commutant

Let $p$ be an odd prime number, let $A\in M_p(\mathbb{F}_p)$ be a $p$-by-$p$ matrix with coefficients in $\mathbb{F}_p$, let $C(A)$ be the commutant of $A$, and let $N\in M_p(\mathbb{F}_p)$ be a ...
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 ...
4 votes
1 answer
193 views

A question on eigenvalue of parametric matrix

Is there a way to efficiently check if all matrices in the following set are Hurwitz stable (eigenvalues strictly in the left-hand plane)?$$\left\{ A \in \Bbb R^{n \times n} : \ell_{i,j}\leq A_{i,j} \...
0 votes
1 answer
235 views

Eigenvalues of $\operatorname{diag}({\bf v}) - {\bf v} {\bf v}^\top - \alpha({\bf v} - {\bf w})({\bf v} - {\bf w})^\top$

Given vectors ${\bf v}, {\bf w} \in [0,1]^n$ , where $n \in \mathbb{N} \setminus \{0\}$, and $\alpha > 0$, I would like to find the eigenvalues of the following matrix. $$\operatorname{diag}({\bf v}...
0 votes
1 answer
134 views

Full matrix ring over an infinite division ring with a finite maximal unital subring?

I'm wondering if there is an infinite division ring $D$ and a finite unital subring $R$ of the full matrix ring $M_n(D)$ ($n$ some positive integer) such that there are no rings properly between $R$ ...
20 votes
2 answers
1k views

Find $Y\in\operatorname{GL}_n(\mathbb{Z})$ such that all eigenvalues of $YX$ are nonnegative

I saw this problem some years ago and I would greatly appreciate any reference or solution. Let $X \in \operatorname{M}_n ( \mathbb{R} )$. Prove that there is $Y \in \operatorname{M}_n ( \mathbb{Z} )$...
1 vote
1 answer
102 views

Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex

Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
1 vote
1 answer
330 views

How to represent infinite matrices in Mathematica?

I asked this question on Mathematica site and the answer was basically "it is impossible". So, some math theory needs to be thrown in besides coding. Let us represent infinite matrices as ...
2 votes
1 answer
131 views

Generate a low-rank sparse covariance matrix

May I ask how to generate a low-rank sparse covariance matrix? Thank you!
2 votes
1 answer
414 views

Integrality certification for product of two matrices $A B^{-1}$

Let's consider two non-singular integer matrices $A,B \in\mathbb{Z}^{n\times n}$. I want a test to check if $A\times B^{-1}$ is integral (or no denominators). I am referring the unimodular ...
3 votes
0 answers
150 views

What are the measure of the volume and boundary (and other quermaß measures) of the positive semidefinite matrices?

Let $E$ be the real vector space of $n\times n$ real symmetric (resp. complex Hermitian) matrices, and $E_1$ those with trace $1$. Endow $E$ with the bilinear (resp. sesquilinear) form given by $(P,Q)...
1 vote
1 answer
332 views

Solvability of $A X B=C$ with $X=X^\mathrm{T}$

I am studying symmetric solutions to the complex matrix equation \begin{equation} A X B=C, \end{equation} where $A$, $B$, and $C$ are $m\times n$, $n \times k$, and $m \times k$ complex matrices, ...
0 votes
0 answers
32 views

What is the impact of individual estimate on each other in matrix inversion?

I am looking to understand the impact of each estimate on each other in matrix inversion. Lets say I have a vector $A = \left[a_1, a_2 \right]^T$ of size $2 \times 1$ and $a_1$ and $a_2$ are related ...
2 votes
1 answer
418 views

Effect of duplicated row on singular values and vectors

Let $\mathbf{A}$ be a $n\times n$ matrix with Singular Value Decomposition (SVD) $\mathbf{A}=\mathbf{U}\mathbf{S}\mathbf{V}$ and $\mathbf{a}_1$ be the first row of $\mathbf{A}$. What can we say about ...
3 votes
1 answer
373 views

Minimum upper bound for sum of the entries of the inverse covariance matrix

Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel $$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$ and let $\mathbf{K}$ be the following $n \times n$ covariance matrix $$\mathbf{K} = \...

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