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Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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1k views

What did Gelfand mean by suggesting to study “Heredity Principle” structures instead of categories?

Israel Gelfand wrote in his remarkable talk "Mathematics as an adequate language (a few remarks)", given at "The Unity of Mathematics" Conference in honor of his 90th birthday, the following ...
32
votes
0answers
2k views

Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
24
votes
0answers
748 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 ...
23
votes
0answers
854 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
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
761 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 \...
17
votes
0answers
325 views

Is it consistent with ZF that $V\to V^{\ast \ast}$ is always surjective?

In a comment to a recent question, Jeremy Rickard asked whether it is consistent with ZF that the map $V \to V^{**}$ from a vector space to its double dual is always surjective. We know that "always ...
16
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0answers
688 views

How to explain the picturesque patterns in François Brunault's matrix?

How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? -- Choosing colors according to the highest power of 2 which ...
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
392 views

Bunnity of multilinear maps

Is there a way to compute the following nullity of multilinear maps? As it is different from any nullity I know of, I call it bunnity after myself:-)) If it already has a name, it be nice to know it. ...
15
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0answers
592 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
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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,\...
13
votes
0answers
182 views

Is the set of power matrices decidable?

Let $\text{Mat}(n\times n,\mathbb{Z})$ denote the collection of integer $n\times n$ matrices. We say $M\in \text{Mat}(n\times n,\mathbb{Z})$ is a power matrix if there is an integer $k>1$ and a ...
13
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0answers
289 views

Cardinality vs. isomorphism type of vector spaces without choice

One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem: If $V$ is an infinite vector space over a field $F$, and $...
13
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0answers
277 views

A determinant problem for primes $p\equiv 1\pmod4$

Let $p$ be an odd prime, and let $A_p$ denote the matrix $$[a_{ij}]_{1\le i,j\le (p-1)/2},$$ where $$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
13
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0answers
417 views

Who stated and proved the “Hopf lemma” on bilinear maps?

If $A\otimes B\rightarrow C$ is a nondegenerate linear map, where $A, B, C$ are vector spaces over an algebraically closed field, then $\dim C\ge \dim A + \dim B -1$. Nondegenerate here means that ...
12
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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
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0answers
189 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
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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 $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
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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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553 views

On a tentative generalization of the Schmidt decomposition

Background I am a PhD student in Physics and I am currently developing quite refined computer codes that allow to simulate many-body quantum systems living on a lattice. The difficulty resides in ...
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 ...
12
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0answers
325 views

Matroids with prescribed independent sets

Let $A$ be a finite set. Let $B$ be a family of subsets of $A$. We are interested in a matroid with a minimum rank such that every element of $B$ is independent. The answer is obvious - a uniform ...
11
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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
votes
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
606 views

An elementary linear algebra problem

Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq V_0$...
11
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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
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0answers
137 views

Cospectral mate of rhombic dodecahedron

I am wondering if the following pair of cospectral graphs was previously known. The rhombic dodecahedron graph looks like this (graph6 string: 'M?????rrAiTOd_YO?'): As far as I know, it was ...
10
votes
0answers
634 views

Dimensions of dual vector spaces

Let $V_F$ be an infinite dimensional right $F$-vector space (over a field $F$, or even over a division ring). The dual space $V^{\ast}={\rm Hom}(V,F)$ is naturally a left $F$-vector space (coming ...
10
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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, ...
9
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0answers
160 views

Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues

Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every ...
9
votes
0answers
233 views

Does there exist an algorithm that decomposes a matrix into a minimal number of elementary matrices for $F_{2}$?

If $i \neq j$, then let $C_{i,j} : F_{2}^{n} \to F_{2}^{n}$ be the elementary linear transformation defined by $$C_{i,j}(x_{1},\dots,x_{i},\dots,x_{j},\dots,x_{n}) :=(x_{1},\dots,x_{i},\dots,x_{i}\...
9
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0answers
357 views

Formula or estimates for $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$

Given two positive definite matrices $A,B$ with nonnegative entries, I seek convenient ways to analytically compute or estimate $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$, where $\...
9
votes
0answers
257 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
421 views

An identity for Hankel determinants

Is the following result about Hankel determinants known or a simple consequence of some known results? Let $f(x) = \frac{\displaystyle 1}{{\displaystyle 1 - \frac{{a x^{m + 2}}}{\displaystyle {1 - \...
8
votes
0answers
363 views

Prove the optimality of the following constant

Let $E$ be a complex Hilbert space. In (arXiv) it was shown that for $A=(A_1,...,A_n) \in \mathcal{B}(E)^n$ we have, $$\displaystyle\frac{1}{2\sqrt{n}}\|A\|\leq \omega(A) \leq \|A\|,$$ where $$ \...
8
votes
0answers
169 views

Is the discriminant of a free (as a module) $R$-algebra always congruent to a square modulo 4?

Let $R$ be a commutative ring. Let $A$ be an $R$-algebra (i.e., an $R$-module equipped with an $R$-bilinear multiplication map that turns $A$ into a unital ring). We do not require $A$ to be ...
8
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0answers
228 views

If $A$ is an algebra, $Sym^n(A)$ is an algebra. Where can I learn more about this algebra structure?

$\newcommand{\Vect}{\mathsf{Vect}} \newcommand{\nats}{\mathbb{N}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\Alg}{\mathsf{Alg}} \newcommand{\CAlg}{\mathsf{CAlg}} \newcommand{\Hom}{\mathrm{Hom}}$ Let ...
8
votes
0answers
271 views

How bad can $SK_1$ of a commutative ring be?

For a commutative ring $R$ define $\mathrm{SK}_1(n, R)=\mathrm{SL}(n, R)/\mathrm{E}(n, R)$, the quotient of the special linear group by its subgroup generated by the elementary matrices. When $n\...
8
votes
0answers
292 views

Conjecture on matrix with reciprocal principal minors

Some notation: $A(\alpha|\beta)$ is the submatrix of $A \in \mathbb{R}^{n \times n}$ with with rows $\alpha$ and columns $\beta$. $\textrm{det } A(\alpha|\alpha) =: \textrm{det } A(\alpha)$ are the ...
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
199 views

Intersection of Springer fibre and Schubert cell

Let us consider intersections of Springer fibres and Schubert cells in type A. Let $ Y : \mathbb C^n \rightarrow \mathbb C^n $ be a nilpotent operator. Let $$ F_Y = \{ V_0 = 0 \subset V_1 \subset \...
8
votes
0answers
215 views

When is a product of hyperbolic matrices hyperbolic?

Suppose $A_1,\ldots,A_n$ is a sequence of $2 \times 2$ complex matrices such that $| \det(A_j) | =1$ and $ | \mathrm{tr}(A_j) | > 2 $ for each $j$. What kinds of reasonable restrictions can one ...
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
471 views

Maximal set on hypersphere that does not contain pairs of orthogonal vectors

Let R be a region on a hypersphere. Each point A of the hypersphere is associated with a vector pointing to A and with origin at the centre of the hypersphere. So let me identify each point with a ...
8
votes
0answers
440 views

integral matrix of order p

Hi everyone Let $p$ be a prime number. I am interested to classify $\{ A\in {\rm GL}_{p-1}(\mathbb{Z}): {\rm ord}(A)=p \}$ up to conjugacy. One reason to consider this problem is its relation to ...
8
votes
0answers
795 views

coordinate-free proof of transitivity of norms or traces

Hello: Suppose $A$ is a finite free $B$-algebra and $B$ is a finite free $C$-algebra. Does anyone know a coordinate-free proof (i.e. without choosing bases) of the identity: $N_{A/C} = N_{B/C}\circ ...