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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.

26
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346 views

Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?

For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
24
votes
0answers
745 views

Real square roots of symmetric matrices

In joint work with Andreas Fischle (TU Dresden, Germany) and Patrizio Neff (U Essen, Germany) we needed to use the following statement: If $S$ is a real $n\times n$ matrix with $S^2$ symmetric, then ...
21
votes
0answers
695 views

Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials

While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
21
votes
0answers
2k views

The Fourier Transform of taking Eigenvalues

The purpose of this question is to ask about the Fourier transform of the map which associate to an $n$ by $n$ matrix its $n$ eigenvalues, or some function of the $n$ eigenvalues. The main motivation ...
20
votes
0answers
4k views

An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?

A famous result in linear algebra is the following. An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$. I know one proof using the Smith Normal Form (SNF). ...
20
votes
0answers
760 views

Cauchy matrices with elementary symmetric polynomials

$\newcommand{\vx}{\mathbf{x}}$ Let $e_k(\vx)$ denote the elementary symmetric polynomial, defined for $k=0,1,\ldots,n$ over a vector $\vx=(x_1,\ldots,x_n)$ by \begin{equation*} e_k(\vx) := \sum_{1 \...
19
votes
0answers
295 views

“quantum” symmetric plane partitions beget alternating sign matrices?

The "quantum" version qTSPP of the number of totally symmetric plane partitions, contained in the cube $[0,n]^3$, is enumerated by $$f_n(q):=\prod_{j=1}^n\prod_{k=1}^j\prod_{\ell=1}^k\frac{1-q^{j+k+\...
16
votes
0answers
619 views

How boundedly generated is $SL_3(\mathbb{Z})$?

The group $G = \mathrm{SL}_3(\mathbb{Z})$ is known to be boundedly generated, that is, there exists some $m \in \mathbb{N}$, and $g_1, \dots, g_m \in G$ such that we have the following equality of ...
15
votes
0answers
223 views

An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically: Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...
15
votes
0answers
478 views

When is the determinant an $8$-th power?

I am working over $\mathbb{R}$ (though most of the story goes over any field). I am looking for linear spaces of matrices such that the restriction of the determinant to this spaces can be written (...
15
votes
0answers
591 views

Determinant inequality involving Hermitian, positive definite matrices

Let $A,B,C\in M_{n}(\mathbb C)$ be Hermitian and positive-definite matrices such that $A+B+C=I_{n}$. Show that $$\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$$ This question has been ...
14
votes
0answers
347 views

The rank of a “triangle-free” matrix

This is a version of the question I asked recently, but the assumptions got now strengthened substantially. Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
14
votes
0answers
664 views

Is “being a full ring of quotients” a Morita invariant property?

Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...
13
votes
0answers
407 views

Descartes rule of signs for a noncommutative polynomial in matrix variables

Recently, while studying certain notions of "averaging" a set of input matrices, I obtained a nonlinear polynomial in matrix variables. A simple example is \begin{equation*} \mathcal{G}(X) := X^n - \...
13
votes
0answers
814 views

Diagonalizing some matrices arising from Fourier transform on $S_n$.

Consider the function $f$ on $S_n$ which equals $1/n$ on all adjacent transpositions $(i,i+1)$, where we let $n+1 = 1$, and $0$ otherwise, and its Fourier transform $\hat{f}(\rho)$ evaluated at the ...
12
votes
0answers
562 views

Pointwise (Hadamard) matrix product and the rank

$\DeclareMathOperator{\rk}{rk}$ Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have $$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...
12
votes
0answers
188 views

Which ordering of factors is needed to obtain this kind of determinantal inequalities?

Let $A$ and $B$ be $n\times n$ Hermitian positive definite matrices. The curious determinantal inequality given here, which can be stated as $$\det (A^{4}+ ABBA+BAAB+B^{4})\ge\det(A^{4}+ AABB+BBAA+B^{...
12
votes
0answers
280 views

Ratio of entries of A and log A where A is a triangular matrix

Consider triangular matrices $A = \left( {a(n,k)} \right)$ of arbitrary order with $a(n,k) = 0$ if $n + k$ is odd and $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
12
votes
0answers
649 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
12
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0answers
644 views

Regular languages of matrices and their generating functions

My question is somewhat related to this question. Let us fix natural numbers $k$ and $C$. Let $A$ be an automaton whose alphabet consists of $k\times k$ matrices with integer coefficients of ...
11
votes
0answers
191 views

Matrices that admit a power that is symmetric

We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
11
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0answers
177 views

Fast computation of matrix product $AXA^T$ with fixed $A$?

Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
11
votes
0answers
251 views

Jaffard's theorem - finite matrices

For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,...
11
votes
0answers
148 views

Is my matrix perturbation analysis legitimate?

I am not a matrix theorist, or numerical linear algebra expert, but I have a problem and my proposed solution leads me to a question that I cannot answer. I can give more details, but the gist is ...
11
votes
0answers
284 views

Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial

This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But.... Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
10
votes
0answers
181 views

What are these matrices called?

A paper I'm writing heavily uses block diagonal matrices with the property that each block is upper triangular and constant along its diagonal. Like this: $$\begin{bmatrix} A_1 \\ & \ddots \\ &...
10
votes
0answers
140 views

Poset of nonvanishing minors of a matrix

This question was posed on MSE here three days ago, but hasn't gotten any answers or suggestions. I hope it's okay to ask it on MO, but if I should wait a little longer, please just let me know. Say $...
10
votes
0answers
187 views

Matrix identities in two variables

The famous Amitsur-Levitzki Theorem states that the algebra $M_n(\mathbb C)$ satisfies no polynomial identity of degree less than $2n$ and it satisfies $$p(x_1,\dots,x_{2n}) = \sum_{\sigma \in \frak ...
10
votes
0answers
206 views

A $k \times n$ matrix with a lot of invertible $k \times k$ submatrices over $\mathbb{F}_2$

In the appendix of the paper by Tolhuizen ( http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=841182) there is a very fast and easy probabilistic proof that for $k=cn$ where $c \in [0,1]$ is fixed, ...
10
votes
0answers
280 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using >this< formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $V= \begin{...
9
votes
0answers
113 views

Comparison of the absolute value of an operator with its positive parts, II

Suppose $A,B\in M_n(\mathbb C)$ are self-adjoint. Does there exist a constant $C>0$ depending only on $n$ such that $$ |A+iB| \leq C(|A| + |B|)? $$ One can take $C=1$ if $A$ and $B$ commute. More ...
9
votes
0answers
256 views

Eigenvalues of leading principal submatrix of the Clement-Kac-Sylvester tridiagonal matrix

It's well-known that the eigenvalues of the Clement-Kac-Sylvester tridiagonal matrix $$\begin{pmatrix} 0 & n-1 & 0 & \dots & 0 \\\ 1 & 0 & n-2 & \dots & 0\\\ 0 & ...
9
votes
0answers
231 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
9
votes
0answers
168 views

Is there any study about positive definiteness of some matrix space whose matrices don't have to be positive-definite?

--Updated description-- I'm trying to investigate the stability of tensegrity structures, and this question is related to the second order test. Suppose there is a vector space $W=\operatorname{span}...
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 ...
8
votes
0answers
171 views

Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
8
votes
0answers
438 views

Name for an operation on matrices?

Given two matrices $A$ and $B$ of size $a \times n$ and $b \times m$ consider the following operation $A \dagger B$ whose result is an $a b^n \times n m$ matrix. $A \dagger B$ is a block matrix with $...
8
votes
0answers
205 views

Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ...
8
votes
0answers
554 views

Bounding sum of first singular values squared for Kronecker sum of traceless matrices

Let $A$ and $B$ be $4\times4$ traceless matrices with Hilbert-Schmidt norms summing up to $1/4$, i.e. $$\text{Tr}\left[ A\right]=\text{Tr}\left[ B\right] = 0,\qquad\text{Tr}\left[ A^\dagger A + B^\...
8
votes
0answers
668 views

Path connected set of matrices?

Consider the collection of $n$ by $n$ matrices $$S=\{ A: A_{ij}\le0,\quad (-1)^{c_i}\det A(P_i;Q_i)<0 \quad \text{for} \quad i=1,\ldots, k\}$$ where $c_i\in \{0,1\}$, $P_i$ and $Q_i$ are disjoint ...
8
votes
0answers
208 views

Standard polynomials applied to matrices (bis)

The standard polynomial in $r$ non-commuting indeterminates $x_1,\ldots,x_r$ is defined by $${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\...
7
votes
0answers
107 views

A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...
7
votes
0answers
99 views

A generalization of matrix minors to non-integer values

I am interested to know if there exist a notion of $k$-minors of a real square matrix, for non-integer positive values of $k$ One approach I thought of was to use the fact that the $k$-minors are (...
7
votes
0answers
147 views

Bound on gap between least eigenvalues of $n \times n$ correlation matrix and of its $(n -1) \times (n-1)$ submatrices

The following problem is motivated by one of my research problems. Let $\Sigma$ be an $n \times n$ correlation matrix whose least eigenvalue is denoted by $\lambda$. $\Sigma_i'$ be an ...
7
votes
0answers
161 views

Counting 0-1 $n\times n$ matrices with a given rank r

What is the number $N$ of $n \times n$ $0$-$1$ matrices with rank $k$? I read this sequence is "OEIS A064230 Triangle $T(n,k)$ = number of rational (0,1) matrices of rank $k$ ($n\ge 0$, $0\le k\le ...
7
votes
0answers
143 views

A special eigenvalue problem

For my research I need to solve a generalised eigenvalue problem $Ax=\lambda B x$, where $A$, $B$ are general matrices, and selectively find only eigen-pairs $\lambda, x$ such that $\lambda\in \mathbb{...
7
votes
0answers
160 views

Quantum coupon collection: positivity of an alternating sum of matrices

It is well-known that in the classic coupon collecting problem (CCP), the expected waiting time is \begin{equation*} T_n(x_1,\ldots,x_n) = \sum_{k=1}^n (-1)^{k+1}\sum_{1\le i_1 < \cdots < i_k \...
7
votes
0answers
217 views

A conjecture of Lubotzky on ranks of subgroups of special linear groups over the integers

In a 1985 paper named "Dimension function for discrete groups" Lubotzky conjectured that: For any integer $n \geq 3$ the group $\mathrm{SL}_n(\mathbb{Z})$ contains infinitely many finite index ...
7
votes
0answers
155 views

Does $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ have periodic points missing the critical hypersurface?

I am trying to prove that if $\varphi: \mathbb{P}^{n} \to \mathbb{P}^{n}$ is an algebraic morphism of degree $d > 1$ (by which I mean $\varphi^{*}(\mathcal{O}(1)) = \mathcal{O}(d)$, so the ...
7
votes
0answers
919 views

Possible values of eigenvalues of Hadamard product of Hermitian matrices

One of the most important (and very well-known) result in the study of the spectrum of Hermitian matrices is Horn's conjecture (or theorem?), which provides a complete answer to the following problem: ...