# Questions tagged [linear-algebra]

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

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### 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**

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**0**answers

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

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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**

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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**

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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**

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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

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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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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

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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**

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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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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**

votes

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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

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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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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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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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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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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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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**

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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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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**

votes

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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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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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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**

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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

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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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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

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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

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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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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**

votes

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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

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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**

votes

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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

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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

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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

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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

**0**answers

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**

votes

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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

**0**answers

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

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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

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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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

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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

**0**answers

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

**0**answers

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

**0**answers

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 ...