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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.

12
votes
1answer
359 views

Is the image of the map $A \to \bigwedge^k A$ closed over $\mathbb{R}$?

Let $V$ a real vector space of dimension $d$. Let $1<k < d-1$. Consider the map induced by the exterior algebra functor: $$ \psi:\text{End}(V) \to \text{End}(\bigwedge^kV) \, \, \, \, , \, \, ...
2
votes
0answers
182 views

How to solve the inverse problem of least-squares?

Focusing on following least squares problem: $$\min\limits_{V} \lVert Z - WV \rVert _{_F}^2$$ $$Z∈{R}^{m*n},\quad W∈{R}^{m*k},\quad V∈{R}^{k*n},\quad k\lt m\lt n $$ This problem can be easily ...
2
votes
1answer
48 views

Local distribution of sample covariance matrix when the number of observations/realisations is less than the matrix dimension

Given a true covariance matrix $M$ of dimension $p \times p$, we generate $n$ gaussian random vectors $X_1,..X_n \sim N(0,M)$. We then get a sample covariance matrix $M_s$ based on these $n$ ...
8
votes
0answers
142 views

Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...
25
votes
2answers
528 views

Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...
2
votes
0answers
155 views

Real-rooted polynomials with coefficient constraints

My question is whether there exists $(a_0, a_1, \ldots, a_{2n-1}) \in \mathbb{R}_{+}^{2n}$ such that (1). $a_{2k} + a_{2k+1} = \binom{3n-1}{3k} + \binom{3n-1}{3k+1} + \binom{3n-1}{3k+2}$ for all $0 \...
1
vote
0answers
33 views

Classification of Maximal Rank Skew-Symmetric Matrices with Linear forms as entries

I am interested in canonical forms/simplifying assumptions of $(2n+1)\times (2n+1)$ skew symmetric matrices with homogeneous degree 1 polynomials in $k[x,y,z]$ as entries, and whose rank is $2n$. I ...
6
votes
2answers
293 views

Probability of a large random integer Matrix to have zero determinant

Suppose we have a matrix $A \in \{0,1\}^{n \times n}$ where $$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad 1-p\end{cases}$$ I would like to ...
1
vote
1answer
200 views

On a condition for a matrix sum to be zero

Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...
1
vote
0answers
85 views

Matrix Sparsity Pattern

Suppose I have a matrix $H^{+} = (H^T H)^{-1} H^T$ where $H$ is a sparse matrix. Consider the case where only the sparsity pattern i.e. the zero elements of $H^{+}$ matrix is known, then would it be ...
4
votes
2answers
144 views

Bound on sum of $n$th super-diagonal entries in a $2n$ by $2n$ PSD matrix

Let $A,B,C\in\mathbb{R}^{n\times n}$ be such that $\left(\begin{array}{} A & B \\ B^T & C \end{array}\right)\succeq 0$. I would like to prove that $$\mathrm{trace}\,B \le \sum_{i=1}^n \sqrt{\...
2
votes
0answers
135 views

The nonlinear operator defined as the commutator of a matrix and a nonlinear operator

In my studies of applied analysis and applied linear algebra, this interesting problem and concept came up: Let us consider the space of all $ m \times n $ real matrices, and define a scalar ...
4
votes
0answers
78 views

Totally Unimodular matrix edited from ordinary matrix

Given a matrix $M\in\{0,1\}^{m\times n}$ is there an algorithm to tell if we can convert some of $1$s to $-1$s and make $M$ Totally Unimodular and output such a Totally Unimodular in polynomial in $mn$...
7
votes
1answer
197 views

Square root of a large sparse symmetric positive definite matrix

I am trying to calculate $$Y = A^{\frac 12} X$$ where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
0
votes
0answers
29 views

transfer a relation from matrices to their cholesky decompositions

I'm stuck on a seemingly easy problem. $$ LL' = TAA'T' $$ a matrix A (lower triangual matrix -> derives from a cholesky decomposition of a covariance matrix -> ( A * A' ) is positive semidefinite and ...
4
votes
0answers
197 views

An upper bound on the Jordan condition number of a matrix

The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
2
votes
1answer
178 views

Solving a “reversed” Stein equation

Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation $$\label{star}\tag{$\star$} XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}. $$ My question. Is it true ...
3
votes
1answer
124 views

Convexity of the matrix mapping $X^{-2}$

Let $X$ be a positive semidefinite matrix. Is the mapping $X\to X^{-2}$ convex? Update: or is $Tr[X^{-2} K]$ convex for PSD $X$ and $K$?
0
votes
1answer
76 views

On the (qualitative) behavior of a coupled differential equation

Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following ...
2
votes
1answer
130 views

Rotatable matrix, its eigenvalues and eigenvectors

We say that a real matrix is rotatable iff after turning it clockwise on $90^{\circ}$ it doesn't change. I'm interesting about eigenvalues and eigenvectors (belonging to non-zero eigenvalues) of such ...
3
votes
0answers
103 views

Representation of a matrix ring

Years ago, I read a paper about how to re-write any $n\times n$ matrix $X$ over the ring $\mathbb Z_N$ (where $N=pq$ for two primes $p$ and $q$) as a product of sequences of two simpler matrices $S$ ...
1
vote
0answers
514 views

On the basis of a finite dimensional vector space (revised)

Revision in response to the comments to earlier version: To introduce the notion of a basis of a finite dimensional vector space over an arbitrary field $\Lambda$, without performing any computation ...
3
votes
0answers
173 views

Diagonal elements of Hermitian matrices with given eigenvalues

Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...
1
vote
0answers
25 views

Depth bound of generating an matrix algebra

Let $X$ be Hilbert spaces $\mathbb{C}^d$, and $L(X)$ be the sets of linear operators of $X$. We are given a matrix subspace $S\subset L(X)$. Via the following procedure, one can generate the smallest ...
0
votes
0answers
111 views

L1 norm constraint on product of 2 matrix

I want to solve below minimization problem \begin{equation*} \begin{aligned} & \underset{A, B}{\text{minimize}} & & ||Y-AB^T -D||_F^2 \\ & \text{subject to} && |A_i|_1 \leq a,...
1
vote
1answer
141 views

Is this totally unimodular family?

Is it possible to prove this matrix family only contains totally unimodular matrices? The matrix has dimensions $\frac{3n(n-1)}2$ rows and $n+\frac{n(n-1)}2$ columns. To every pair $(i,i')$ with $1\...
5
votes
2answers
190 views

Factoring a positive semidefinite matrix into binary matrices

This question is motivated by a research problem I recently encountered. Consider two sets of random variables $\mathbf{X}$ and $\mathbf{Y}$, where $\mathbf{Y}$ can be expressed as a linear ...
4
votes
0answers
106 views

Maximizing a certain eigenvalue ratio

Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following ...
4
votes
1answer
254 views

A generalization of unsolvable equation $ab-ba=1$ in a Banach algebra

It is well known that the equation $$(*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra. I search for some reasonable generalization of this equation in higher variable for investigation ...
0
votes
0answers
29 views

Solutions to this equation of the form $A(t_1,t_2)x = b(t_2)$

Given two symmetric positive definite matrices $L_1, L_2\in\mathbb{R}^{n\times n}$ and a vector $w\in\mathbb{R}^n$, under what conditions does the following system of equations of the form $M_1(t_1,...
1
vote
2answers
88 views

Solution of $\dfrac{d [\epsilon_i]}{dt} = (\beta \mathbf{A} -\delta \mathbf{I})\left[\epsilon_i\right] - \alpha \mathbf{B} \left[\epsilon_i^2\right]$

I have a problem that can be described by the following equation: \begin{equation} \dfrac{d \left[\epsilon_i\right]}{dt} = \left( \beta \mathbf{A} - \delta \mathbf{I} \right) \left[\epsilon_i\right] -...
5
votes
1answer
356 views

higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
1
vote
0answers
114 views

identity involving spectral functions

Let $A$ be any compact operator and let $A^*$ denote its adjoint. Let $f$ be a spectral function. Then is the following true : $$ A^* f(AA^*) = f(A^* A) A^*$$
0
votes
1answer
127 views

Searching for matrices with some property

I don't know if this question is considered research-related. If not, I will move it to Math SE. I am searching for matrices with the property $$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) ...
2
votes
0answers
127 views

Eigenvalues of special sum of Hermitian matrices

In my research on linear algebra and its applications, I have come across the following problem which has stumped me: Let $ A $ be a positive definite matrix and let $ D $ be a positive diagonal ...
1
vote
1answer
92 views

Rank of matrix n x m [closed]

What is the most efficient way to calculate the rank of matrix A with dimmension n x m? I am intrested if it's rank is highest possible -> rank(A) = n Thank you in advance
0
votes
0answers
80 views

The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$

The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$. Let algebraic connectivity of $P_n$ be denoted by $\mu$. I have proved a result that if $G$ is a ...
-2
votes
1answer
52 views

Find a columns of matrix $A$ which form a basis of columns space of matrix $A$ [closed]

We have a matrix $A$ whose rows are data records and whose columns are features. We would like to omit useless features such as zero or constant columns, duplicate columns, columns that are equal to ...
2
votes
2answers
365 views

Finding an adjacency matrix whose cube's diagonal is equal to a given vector

How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and $$\mbox{diag}\left(A^3\right)=b$$ for some given vector $b$? Is there a way to characterize all ...
1
vote
1answer
84 views

SVD of two matrices A and B having the same right singular vectors?

I saw this statement in a lecture note Assume the generalized SVD of matrices $A\in R^{m\times n}$ and $B\in R^{p\times n}$ given as: $$U^TAX = diag(\alpha_1, ..., \alpha_n),~ U^TU = I_m$$ $$...
0
votes
0answers
113 views

Symmetric Grothendieck inequality

Grothendieck's inequality states that for all $n \times n$ matrices $(a_{ij})$ such that $$\max_{x \in \{\pm 1\}^n,\, y \in \{\pm 1\}^n} \left|\sum_{ij} a_{ij}\, x_i\, y_j\right| \leq 1,$$ there ...
5
votes
2answers
579 views

Nuclear norm as minimum of Frobenius norm product

Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix. It is claimed that $$ \|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
8
votes
1answer
252 views

Distance from nonnegativity of some orthonormal vectors

Suppose that $1 < k < n$. Does there exist a constant $\beta > 0$, such that for every $k$ orthonormal vectors $f_1,\ldots,f_k \in \mathbb R^n$, there exist $k$ orthonormal vectors with ...
9
votes
2answers
506 views

Do two dimensional representations with the same adjoint representation differ by a character?

Let $K$ be a field of characteristic not equal to $2$. Let $\text{ad} : \text{GL}_2(K) \to \text{GL}_3(K)$ be the adjoint representation, obtained by $\text{GL}_2(K)$ acting on $2 \times 2$ matrices ...
11
votes
2answers
546 views

Inverse of a small submatrix

Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
1
vote
1answer
115 views

Summation unknown notation [closed]

I am having trouble with a summation notation in a paper I am reading (talking about Semi Markov Processes), I am not how to use it. The equation is as follow: $$MTTSF_{\phi}=\sum_{i - 1}\frac{1}{\...
2
votes
1answer
117 views

An inequality regarding projection

Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define ...
1
vote
1answer
101 views

Upper bound on the number of non-zero entries of the product of sparse matrices

I have two sparse matrices: $A$ of dimension $m \times k$ and $B$ of dimension $k \times n$. Is there a way to know how many non-zero entries there are in $C = A B$ without computing $A B$? I can ...
2
votes
1answer
77 views

Lower bound for rank of a matrix

This was asked earlier at MSE and incorporates replies from Omnomnomnom and Chappers. Let A = (a$_{i,j}$) be an $\,$ n x n $\,$ real symmetric matrix. What can be said about lower bounds for rank(...
0
votes
1answer
78 views

Limit of eigenvalues of a matrix perturbation sequence

Suppose $H$ is an $n\times n$ symmetric positive definite matrix, $M_k$ is a sequence of $n \times n$ matrix (not necessarily symmetric) such that $M_k \to O$ where $O$ is the zero matrix. Let $\...