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21 votes
1 answer
653 views

Characteristic polynomial of the Gcd matrix

Let $A_n$ be the $n \times n$-matrix with entries $\gcd(i,j)$ and $f_n$ the characteristic polynomial of $A_n$. Question: Is $f_n$ irreducible over $\mathbb{Q}$ for all $n$ except $n=8$? This is ...
Mare's user avatar
  • 26.5k
10 votes
1 answer
520 views

Homogeneous polynomials, mixed determinants, positive definiteness

Are there $n\times n$ real matrices $A_{1}, \ldots, A_{n}$ such that the $n$-homogeneous polynomial $$ f(x_{1}, \ldots, x_{n}) = \det(x_{1} A_{1}+\cdots +x_{n} A_{n}) $$ never vanishes on $\...
Paata Ivanishvili's user avatar
9 votes
1 answer
307 views

a generalization of gamma matrices

Is it possible to find matrix solutions to the following : $$\left(\sum_1^m M_k x_k\right)^n=\left(\sum_1^m x_k^n\right)I_d$$ where $M_k$ are the desired $d \times d$ matrices (no restriction on $d$) ...
unknown's user avatar
  • 451
8 votes
3 answers
1k views

Relating a Polynomial equation to the characteristic equation of a Hermitian matrix

This question arose out of mere curiosity. Given a polynomial equation and I happen to know that its roots are real (but not the roots itself). Does it mean it is the characteristic equation of a ...
dineshdileep's user avatar
  • 1,421
7 votes
1 answer
248 views

The determinant of a $4\times4$ matrix associated to some specific polynomial as follow

Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by $$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$ where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$. ...
user173856's user avatar
  • 1,997
5 votes
1 answer
418 views

Which positive definite symmetric matrices have solvable characteristic polynomial?

I am interested in the structure of the space of $n \times n$ positive definite symmetric matrices with rational entries whose characteristic polynomials are solvable (i.e. the Galois group is ...
Paul Siegel's user avatar
  • 29.2k
5 votes
1 answer
2k views

annihilator/common left multiple of matrix polynomials

Let $A_{n,d}$ be the space of polynomials of degree $d$ or less whose coefficients are real $n\times n$ matrices --- or, if you prefer, the space of matrices whose entries are degree-$d$ polynomials. ...
Federico Poloni's user avatar
5 votes
1 answer
155 views

Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$ A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}. $$ I'm interested in finding a ...
Ludwig's user avatar
  • 2,712
5 votes
0 answers
315 views

Is there a matrix with this specific quadratic determinant?

We have $\det M=(a+b)(c+d)$ where $M=\begin{bmatrix} a& 0& -1& 0\\ 0& c& 0& -1\\ b& 0& 1& 0\\ 0& d& 0& 1 \end{bmatrix}$ and $\det M'=(a'+b')(c'+d')$...
Turbo's user avatar
  • 13.9k
4 votes
1 answer
127 views

On the linear factors of a polynomial obtained from the determinant of a matrix whose entries are related to Binomial expansion

Consider the polynomial ring $R=\mathbb C[x,y]$. Consider the matrix $A=\begin{pmatrix} x^5+y^5&5x^5&10x^5&10x^5&5x^5\\5y^5&x^5+y^5 &5x^5&10x^5&10x^5 \\10y^5&5y^5&...
user521337's user avatar
  • 1,209
4 votes
1 answer
307 views

How can we find a monic polynomial with the smallest degree in left ideal of $\mathrm{Mat}(F[x])$?

Let $F$ be a finite field, $R=F[x]$ be a polynomial ring and $K = \mathrm{Mat}_n(R)$ be a full matrix ring over $R$. We identify the ring $K$ with the ring $\mathrm{Mat}_n(F)[x]$, for example $$ \left(...
Mikhail Goltvanitsa's user avatar
3 votes
2 answers
450 views

Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture

Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
Alexander Chervov's user avatar
3 votes
1 answer
156 views

How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?

Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$. I would like to show that $$ \{ \mathbf{x} \in \mathbb{A}^n_{\...
Johnny T.'s user avatar
  • 3,625
3 votes
1 answer
102 views

Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)

Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$. For monics $...
Arrow's user avatar
  • 10.5k
3 votes
1 answer
829 views

polynomial matrices and its spectrum

Hello, all! I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$, $$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g _{0,l-1}(...
3 votes
0 answers
295 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
Puzzled's user avatar
  • 8,998
3 votes
0 answers
89 views

The rank of a special matrix

Suppose that $P$ is a polynomial of degree $d:=\deg P$ over a field $\mathbb F$ of zero characteristic, splitting completely into pairwise distinct linear factors, and $B,C\subset\mathbb F$ are sets ...
Seva's user avatar
  • 23k
3 votes
0 answers
419 views

(Expected) Size of smallest singular value of a Vandermonde matrix associated to roots of polynomial

Let $n,H$ two fixed positive integers. Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...
user70925's user avatar
  • 313
2 votes
2 answers
561 views

The number of solutions of a matrix equation

Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ be a polynomial, $a_i \in \mathbb{R}$ for all $i$. Set $$S = \lbrace A \in \mathbb{M}_n: P(A) = 0 \rbrace.$$ We consider the following relation $\sim$ on $S$:...
Pham Hung Quy's user avatar
2 votes
1 answer
331 views

Symmetric polynomials preserving $-1,1$ matrices

If $A$ is an $n\times n$ integer matrix, then trivially $S=A+A^t$ and $P = AA^t$ where $t$ is ``transpose", are both symmetric. Assume that $A$ is also a "$\lbrace -1,1 \rbrace$" matrix, i.e., the ...
Luis H Gallardo's user avatar
2 votes
0 answers
97 views

How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
zxcv's user avatar
  • 131
2 votes
2 answers
492 views

on existence of matrices X, Y s.t. XAY is diagonal over non-commutative ring

Given $A\in Mat_{n\times n}(R)$ where $R$ is a non-commutative associative ring are there exist any (non-zero) matrices $X, Y\in Mat_{n\times n}(R)$ such that $XAY=diag(a_1, \ldots , a_n)$ for some $...
zroslav's user avatar
  • 1,422
1 vote
1 answer
157 views

Global polynomial basis for the kernel of a matrix polynomial

Let $M(x)$ be an $m$ by $n$ matrix with entries in $\mathbb{C}[x]$. Suppose that for all $x\in \mathbb{C}$ the rank of $M(x)$ is constant and equal to $r<n$. Therefore, for any $x_0\in \mathbb{C}$ ...
Peter Kravchuk's user avatar
1 vote
1 answer
141 views

Minimal number of linearly dependent rank-1 projectors

What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent. PS: the ...
Alm's user avatar
  • 1,207
1 vote
0 answers
45 views

Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations

Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
user31127's user avatar
1 vote
0 answers
72 views

Eigenvalues of a subset of matrix semigroup

My apologies for slightly longer post but I wanted to explain lower dimensional cases and their proofs before asking the actual question, which starts after the phrase The general case below. A two-...
Maulik's user avatar
  • 111
1 vote
0 answers
39 views

Characterisation of Coxeter matrices with all non-real eigenvalues having absolute value 1

Let $C$ be an invertible integer matrix. Then a matrix $M$ is called Coxeter matrix (following Sato in https://www.sciencedirect.com/science/article/pii/S0024379505001709?via%3Dihub ) when $M=-C^{-1} ...
Mare's user avatar
  • 26.5k
1 vote
0 answers
188 views

Phase angles of a complex eigenvector

I have the following system for $\lambda \in \Bbb C, \lambda \neq 0$ and $\pmb{p},\pmb{q} \in \Bbb C^n$, $(\pmb{p}^T, \pmb{q}^T)\neq0$: $$\begin{cases} F(\lambda) \pmb{p} - g(\lambda) \pmb{q} - \...
Andrey Gorbunov's user avatar
1 vote
0 answers
95 views

Is it true that the generator of maximal ideal in $M_n(P[x])$ can be choosen to be monic?

Let $P$ be a finite field and $R=M_n(P[x])$ be a matrix polynomial ring. I want to prove that for every polynomial (not necessary with invertible leading term) $A(x)\in R$ such that $R\cdot A(x)$ is ...
Mikhail Goltvanitsa's user avatar
1 vote
0 answers
114 views

Intersecting vector spaces defined over different fields

Let $K_1, K_2$ be subfields of $K$, let $k = K_1 \bigcap K_2$, let $V_1$ be a $K_1$-vector space, $V_2$ be a $K_2$-vector space, both of them subsets of a $K$-vector space $V$. How can I compute a $k$...
Mark's user avatar
  • 314
1 vote
0 answers
137 views

Boundary of pseudospectra

Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x $ is a complex ...
R.T MAN's user avatar
  • 151
1 vote
0 answers
53 views

Distributing partially known data between n parties

Assume that $n = 2r+1$. There are $n$ elements $a_1,a_2,\ldots,a_n$ from a finite field $\mathcal{F}$, and $n$ parties. Each party knows the values of at least $r+1$ elements out of those $n$ elements....
real's user avatar
  • 323
0 votes
1 answer
294 views

Standard rational functions from matrices

In linear algebra we get introduced to standard polynomials that are associated to matrices such as characteristic polynomials and determinants. What are some of the standard rational functions that ...
Turbo's user avatar
  • 13.9k
0 votes
1 answer
141 views

Vandermonde matrix with polynomials

Let us consider the simple Vandermonde matrix $V_n$ with $V_{ij} = \omega^{(i - 1)(j - 1)}$ where $\omega = e^{2i\pi/n}$. Its well known that for a column vector $A$, $VA$ is equivalent to evaluating ...
Aditya Jain's user avatar
0 votes
0 answers
79 views

Quick calculation of a symmetric product with two indices

Say I have a product $\prod_{1\le i \le N-1}\prod_{i<j\le N-1} (1+t_i t_j a_{ij})$, where $a_{ij}$s are real number. I want to calculate the coefficient of $\prod_{0 \le i < N} t_i$. Is there an ...
pallab1234's user avatar