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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simpler question from sylvesters eqn

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 ...
yoyoq's user avatar
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2 votes
0 answers
43 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 $d_0$ ...
Rellek's user avatar
  • 401
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 ...
ElectricPhysiscist's user avatar
0 votes
1 answer
71 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 ...
wlad's user avatar
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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 ...
António Borges Santos's user avatar
-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, ...
rooted-and-grounded's user avatar
3 votes
0 answers
46 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 $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
lntk's user avatar
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1 vote
2 answers
111 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 $\|\cdot\|$ ...
KatsanikJr's user avatar
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 ...
Dragnovith's user avatar
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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)...
bing's user avatar
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3 votes
0 answers
72 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 $j$. ...
user133281's user avatar
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 $B$...
David Bevan's user avatar
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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\...
tony's user avatar
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2 votes
1 answer
249 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 ...
muddy's user avatar
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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} \...
Tardis's user avatar
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-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 $\forall x$...
Shoeb's user avatar
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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 ...
Siddharth Iyer's user avatar
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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 ...
pallab1234's user avatar
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 $...
user61388's user avatar
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 ...
loup blanc's user avatar
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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.
Hamed Elwarfalli's user avatar
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)...
Gro-Tsen's user avatar
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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 ...
Sagar's user avatar
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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\...
Semen Podkorytov's user avatar
1 vote
0 answers
50 views

Ideals of Laurent polynomial ring over matrix ring

Let $K$ be a field. Let $R=M_2(K)\langle x,x^{-1}\rangle$ be the ring obtained from the matrix ring $M_2(K)$ by adjoining two elements $x$ and $x^{-1}$ which are inverse to each other ($x$ and $x^{-1}$...
Ralle's user avatar
  • 471
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 ...
Zach's user avatar
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3 votes
1 answer
102 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 ...
sagnik chakraborty's user avatar
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, ...
Juan's user avatar
  • 61
51 votes
8 answers
5k views

Is there a fast way to check if a matrix has any small eigenvalues?

I have hundreds of millions of symmetric 0/1-matrices of moderate size (say 20x20 to 30x30) which (obviously) have real eigenvalues. I wish to extract from this list the tiny number of matrices that ...
Gordon Royle's user avatar
  • 12.3k
1 vote
1 answer
108 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
0 votes
0 answers
155 views

Simultaneous triangulation and Jordan normal form of commuting nilpotent matrices

Let $A_1,\ldots,A_r$ be $n\times n$ nilpotent matrices over $\mathbb C$, the field of complex numbers, satisfying $A_i\cdot A_j=A_j\cdot A_i$ for all $i,j$. As the matrices commute, they admit ...
sagnik chakraborty's user avatar
3 votes
2 answers
272 views

An analogue of Jacobi's formula for the matrix permanent

Is there an analogoue to Jacobi's formula for the matrix permanent?
Sela Fried's user avatar
2 votes
1 answer
200 views

Is matrix B obtained from matrix A?

Assuming a matrix $\mathbf{A} \in \mathbb{R}^{4096 \times 4096}$ sampled from a standard normal distribution $N(0, 1)$, and another matrix $\mathbf{B} \in \mathbb{R}^{4096 \times 4096}$ either sampled ...
eternity's user avatar
1 vote
0 answers
122 views

Gelfand's representation on matrices: construct maximal ideal in matrix algebra

I would like to see a constructive proof (some algorithm?) of the following statement: Let $A_1, A_2, \dotsc ,A_k \in M_n(\mathbb C)$ be some commuting matrices, let $B$ be the commutative algebra (...
Zhang Yuhan's user avatar
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} := ...
Alex Damian's user avatar
0 votes
1 answer
105 views

Matrix-order derivatives (differentiating a function a matrix number of times)

I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
charlesalexanderlee's user avatar
5 votes
1 answer
78 views

Interpolation between two matrices so that $L^p$ norm is controlled

Assume to have two square matrices $A$ and $B$ acting on $\mathbb{R}^n$ such that all their entries are in the interval $[0,1]$ and such that $||Ax||_1 = ||x||_1$ and $||Bx||_\infty \leq ||x||_\infty$....
tommy1996q's user avatar
0 votes
1 answer
391 views

What is the mathematician's definition of the determinant? [closed]

I am trying really hard to find a good definition of the determinant. I have looked virtually every single resource online and everybody gives a different answer: sum of cofactors or minors https://...
Olórin's user avatar
  • 179
6 votes
1 answer
175 views

Efficiently solve the Sylvester equation $AX+XA = C$ where $X$ is skew-symmetric

Is there a way (more efficient than the standard vectorization) to solve the following Sylvester equation in the skew-symmetric matrix $X$ $$AX+XA = C$$ where the matrix $A$ is symmetric positive ...
Gabi's user avatar
  • 163
2 votes
0 answers
248 views

Two questions about three circulant matrices

Consider the following matrix equation in $n \times n$ circulant $\pm 1$ matrices $A$, $B$, $C$ $$2AA^T+BB^T+CC^T=(4n+4)I-4J$$ where $I$ is the $n \times n$ identity matrix and $J$ is the $n×n$ matrix ...
user369335's user avatar
6 votes
2 answers
626 views

Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?

I have a matrix that is “kind of symmetric.” Specifically, it is an $n \times n$ real matrix such that the entries $a_{ij}=1/a_{ji}$ whenever $j \ne i$. I want to investigate the properties of this ...
bryceadam1's user avatar
0 votes
0 answers
46 views

Conditions on symmetric $3 \times 3$ matrices to satisfy the convex equality for cofactor and determinant

Given any $3\times 3$ finite set of symmetric matrices $A_i$ and positive real $a_i$ such that $\sum_ia_i=1.$ Is there any equivalent condition to the existence of skew symmetric matrices $X_i$ such ...
user519646'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
6 votes
0 answers
145 views

Expressing an invertible sparse matrix as a product of few elementary matrices

Let $M$ be an $n \times n$ matrix with integer entries. Suppose that $M$ is invertible (over the integers) and that $M$ has at most $An$ nonzero entries, each of which is less than $B$ in absolute ...
John Pardon's user avatar
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0 votes
0 answers
50 views

Symmetric indefinite matrix of fixed rank — manifold structure?

I have been studying symmetric indefinite matrices of fixed rank, which have been rather useful for a particular application. I wonder if there is a way to parameterise these by a smooth manifold, e.g....
turtlesandwich's user avatar
0 votes
1 answer
84 views

Change of the smallest positive eigenvalue after a rank-one update

Given natural numbers $n,r,R\in\mathbb{N}$ with $r,R\le n$, let $A\in\mathbb{R}^{n\times r}$ and $B\in \mathbb{R}^{n\times R}$ be two matrices with full column rank and let $c\in\mathbb{R}^n$. Denote ...
Philipp Trunschke's user avatar
0 votes
0 answers
30 views

Elliptic operators with Robin boundary conditions

Can it be proved that two elliptic operators with Robin boundary conditions generate an interval $P$-matrix? $$ -a\Delta u_i = f_i, \quad a\frac{\partial u_i}{\partial n} + bu_i = 0 $$ $$ -a\Delta v_i ...
jokersobak's user avatar
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
3 votes
1 answer
141 views

Does this matrix equation always have a solution?

Let $\{A_i, i\ge 3\}$ be the matrices whose columns represent numbers from $0$ to $2^i-1$ in binary form. For example, $A_3 = \begin{bmatrix} 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \...
Arnaud Casteigts's user avatar
1 vote
1 answer
42 views

Wold decomposition of toral endomorphisms

Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...
an_ordinary_mathematician's user avatar

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