All Questions
35 questions
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 ...
- 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 $\...
- 3,946
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 $d$) ...
- 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 ...
- 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 $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 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 ...
- 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. ...
- 20.2k
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 ...
- 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 $\det M'=(a'+b')(c'+d')$...
- 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&...
- 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
$$
\left(...
3
votes
2
answers
449
views
Guess the next polynoms in the sequence (MO vs. AI :), count anticommuting $F_p$-matrices, P. Hrubeš conjecture
Here is a sequence of polynoms - (presumably) counting N-tuples of ANTI-commuting 2x2 matrices over $F_p, p>2$. (That is just the case of 2x2 matrices, and (surprisingly) it is not so easy to see a ...
- 24.8k
3
votes
1
answer
156
views
How can I show $\{\mathbf{x}: \dim (\ker M_1(\mathbf{x}) \cap \ker M_2(\mathbf{x})) \geq C \}$ is an affine variety?
Let $M_1(\mathbf{x})$ and $M_2(\mathbf{x})$ be $m$ by $m$ matrices with each entry a homogeneous form in $\mathbb{C}[x_1, \ldots, x_n]$.
I would like to show that
$$
\{ \mathbf{x} \in \mathbb{A}^n_{\...
- 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 $...
- 10.5k
3
votes
1
answer
829
views
polynomial matrices and its spectrum
Hello, all!
I have a polynomial non-singular square matrix over $\mathbf{F} _q[x]$,
$$\underset{l \times l}{G(x)} = \left( \begin{matrix} g _{0,0}(x) & g _{0,1}(x) & \ldots & g _{0,l-1}(...
3
votes
0
answers
295
views
Decomposition of a determinant
Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$.
Does there exist a ...
- 8,998
3
votes
0
answers
89
views
The rank of a special matrix
Suppose that $P$ is a polynomial of degree $d:=\deg P$ over a field $\mathbb F$ of zero characteristic, splitting completely into pairwise distinct linear factors, and $B,C\subset\mathbb F$ are sets ...
- 23k
3
votes
0
answers
419
views
(Expected) Size of smallest singular value of a Vandermonde matrix associated to roots of polynomial
Let $n,H$ two fixed positive integers.
Let $P\in\mathbb{Z}[X]$ a monic integral polynomial of height $H$ and degree $n$ taken uniformly at random (i.e. each of the $n$ free coefficients of $P$ is ...
- 313
2
votes
2
answers
561
views
The number of solutions of a matrix equation
Let $P(X) = a_nX^n + \cdots + a_1X + a_0$ be a polynomial, $a_i \in \mathbb{R}$ for all $i$. Set
$$S = \lbrace A \in \mathbb{M}_n: P(A) = 0 \rbrace.$$
We consider the following relation $\sim$ on $S$:...
- 2,773
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 ...
- 1,641
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 ...
- 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 $...
- 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}$ ...
- 1,488
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 ...
- 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 ...
- 21
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-...
- 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} ...
- 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} - \...
- 111
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 ...
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 $k$...
- 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 ...
- 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....
- 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 ...
- 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 ...
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 ...
- 397