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Metrics on the group of unimodular polynomial matrices

The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. ...
Ludwig's user avatar
  • 2,712
2 votes
0 answers
69 views

Tight upper bound for the degree of the entries of adjugate of polynomial matrix

(This question was originally asked at Math.SE, where it didn't receive any answers.) Let $A(x_1, \ldots, x_m)$ be a $n$ x $n$ matrix whose entries are polynomials on real variables $x_1, \ldots, ...
Tadashi's user avatar
  • 1,590
11 votes
1 answer
633 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
Fred Hucht's user avatar
  • 3,671
4 votes
0 answers
208 views

An operator derived from the divided difference operator $\partial_{w_0}$

Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by ...
user173856's user avatar
  • 1,997
2 votes
0 answers
193 views

Hilbert series of the weight 0 sub-algebra of the algebra of functions on GL(N)

Let $A$ be the algebra of polynomials in $N^2$ variables $x^i_j$, $i,j=1,\dots,N$. It is $\mathbb{Z}^N$ graded, with $\text{weight}(x^i_j)=e_i-e_j$. Here $(e_1,\dots,e_n)$ is the standard basis of $\...
Alex Ogg's user avatar
  • 169
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
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
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
0 answers
196 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 ...
Alon Levy's user avatar
  • 113
2 votes
1 answer
222 views

Trace of a Product of Finitely Many Matrices with Cosine Entry

Can someone help me prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} 2\cos\frac{2j\pi}{n} & -m \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} 2 & ...
David Sun's user avatar
  • 309
9 votes
3 answers
544 views

Product of a Finite Number of Matrices Related to Roots of Unity

Does anyone have an idea how to prove the following identity? $$ \mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ...
David Sun's user avatar
  • 309
3 votes
0 answers
113 views

Rank of a particular matrix

Denote $P_{2n}$ to be collection of homogeneous total degree $2$ real polynomials in exactly $2n$ variables such that coefficient of every monomial is either $1$ or $-1$. Split variable set into ...
Turbo's user avatar
  • 13.9k
7 votes
1 answer
484 views

Represent matrix immanants using Schur functions

For each irreducible character $\chi^\lambda$ of the symmetric group $S_n$, the immanant of an $n\times n$ square matrix $A$ is defined as \begin{equation*} d_\lambda(A) := \sum_{\sigma \in S_n} \...
Suvrit's user avatar
  • 28.6k
19 votes
2 answers
5k views

Why is a matrix pencil called a pencil?

I'm trying to understand the historical context behind the word pencil in matrix pencils, or pencil of curves so on. I am aware that even Gantmacher 1959 has this terminology however I don't know ...
percusse's user avatar
  • 295
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
8 votes
7 answers
3k views

Source for roots of matrix polynomials?

A matrix polynomial is a polynomial whose variables are square $n \times n$ matrices, let's say with entries in $\mathbb{C}$, and with coefficients in $\mathbb{C}$. I am seeking a source of results on ...
Joseph O'Rourke's user avatar
4 votes
3 answers
743 views

The Poisson-kernel in the plane and polynomials

Let \begin{align*} p(z) & = \ \prod_{j = 1}^{n} (z - z_j) \, ; \ |z_j| \ = \ 1 \\ & = \ \prod_{j = 1}^{l+1} (z - z_j)^{M_j} \end{align*} be a non-constant complex polynomial with $l+1$ ...
thomashennecke's user avatar
5 votes
1 answer
419 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
24 votes
6 answers
2k views

Cayley-Hamilton revisited

Let $(A_i)_i$ be $n\times n$ matrices with entries in a field $K$ with characteristic $0$. We consider the equation (1) $f(X)=A_kX^k+\cdots+A_1X+A_0=0_n$ where $X\in\mathcal{M}_n(K)$ is unknown. Let $...
loup blanc's user avatar
  • 3,741
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
13 votes
0 answers
458 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 - \...
Suvrit's user avatar
  • 28.6k
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
3 answers
672 views

Do approximately the same polynomials have approximately the same roots? [closed]

"If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to prove ...
Murat Güngör's user avatar
2 votes
1 answer
856 views

Simple and general relation between continuant polynomials

Continued fraction $[a_0,a_1,...,a_n]$ may be expressed as quotient of two polynomials of $(a_0,a_1,...,a_n)$, named continuants (see http://en.wikipedia.org/wiki/Continuant_%28mathematics%29 ) $[a_0,...
kakaz's user avatar
  • 1,626
5 votes
0 answers
1k views

Characteristic polynomial of a symmetric integer matrix

I am wondering what, if anything, is known about the characteristic polynomials of integer symmetric matrices. I believe I read somewhere that not every polynomial with integer coefficients can be a ...
Elena's user avatar
  • 315
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}(...
4 votes
3 answers
1k views

Detecting if a polynomial is a Pfaffian

Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries? The pfaffian can be defined as $\sqrt{{\rm det}(A) } $ when $A$ ...
IMeasy's user avatar
  • 3,779
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
4 votes
4 answers
4k views

Irreducibility of determinant of symmetric matrix

It is quite known fact that the determinant of arbitrary symmetric matrix is an irreducible polynomial in algebra $\mathbb C [x_{ij}, 1\leq i,j\leq n]$ ($x_{ij}=x_{ji}$) (see this: atlas.mat.ub.es/...
zroslav's user avatar
  • 1,422
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
20 votes
2 answers
8k views

Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters

Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
Cristi Stoica's user avatar
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
7 votes
4 answers
526 views

If the series Σ pᵃ⁽ʷ⁾·xᴵʷᴵ is rational, is Σ a(w)·xᴵʷᴵ also rational (summation over words w in a regular language)?

Let $p$ be a prime number and let $a_i$ be a sequence of natural numbers such that the series $\sum_{i=1}^\infty p^{a_i} x^i$ is rational. A warm-up question: Question 1. Does it follow that the ...
Łukasz Grabowski's user avatar
29 votes
12 answers
6k views

When does 'positive' imply 'sum of squares'?

Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares? Example. A positive integer does not ...

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