# Questions tagged [linear-algebra]

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

1,347 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
2k 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 ...
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 ...
854 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 ...
921 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 ...
6k 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). ...
834 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 \...
474 views

### Fundamental Theorem of Algebra via multiple integrals

Consider the product of complex linear monic polynomials times polynomials of degree less than $n$, that is $\big( (z-\lambda), p(z)\big)\mapsto (z-\lambda)p(z)$. If we represent a polynomial by its ...
434 views

293 views

### Combinatorial proof of invertibility of a symmetric matrix associated to the ring of matrices over a finite field

Let $F$ be a finite field of $q$ elements with characteristic $p$. Let $M_n(F)$ be the ring of $n\times n$ matrices over $F$. We define a $q^{n^2}\times q^{n^2}$ symmetric matrix $L$ over the ...
234 views

### Can one Gershgorin circle (only) contain all eigenvalues, when the other circles are not contained in it

In short, following a question from my students, I am trying to find a special case where all the eigenvalues of a matrix lie within only one circle, but not in the others, and the other circles are ...
271 views

### How many steps on $S_n$ are required to span $V\wedge V$, $V = K^n$?

Let $A$ be a set of generators of $G=S_n$; assume $e\in A$, $A=A^{-1}$. Let $V = K^n$, $K$ a field. Consider the natural action of $G$ on $V$ (namely, $g(e_i) = e_{g(i)}$) and on $W = V\wedge V$ (...
747 views

290 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 $a(n,n - 2k) = \frac{{n!}}{{k!(n - 2k)!}}\frac{{(m + n - k - 1)!}}{{(m + n - 1)!}}$...
741 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 ...
575 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 ...
339 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 ...
338 views

### Cohomology for distributive lattices

(edit: Here is a PDF with 5 examples of distributive lattices $L$ with the grades of every point of $L$: https://docdro.id/cUIOb2T . Another class of examples are the divisor lattice where the grade ...