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4 votes
0 answers
167 views

How to prove the following equation (which involves binomials and determinant of 2×2 matrices)?

I have tried many ways to prove the following equation, such as the method of induction and expanding all the terms in the summation,but things got more complicated.I could not find an appropriate ...
tongjun's user avatar
  • 41
2 votes
0 answers
113 views

Numbers of positive terms in polynomials equal A069999

Let $a(n)$ be A069999 (i.e., number of possible dimensions for commutators of $n \times n$ matrices; it is independent of the field). OEIS states that no generating function is known. Let $P(n,k)$ be ...
Notamathematician's user avatar
2 votes
0 answers
117 views

A very specific quotient of a determinantal variety

I'm interesting in knowing whether a certain variety defined by maximal minors is irreducible. The specific construction is as follows: let $n \geq 2$ and let $R = \mathbb{C}[a_1,b_1,c_1,d_1,e_1,f_1,...
Jon Elmer's user avatar
  • 185
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
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 ...
Alexander Chervov's user avatar
0 votes
0 answers
37 views

Largest root of the Adjacency matrix of two graphs (comparison)

Let $G$ and $H$ be two graphs whose spectral radius (largest eigenvalue) of the adjacency matrix is the largest root of the following polynomial: $$P_G(x) = x^6-x^5-(2a-n+5)x^4+(2a-n+1)x^3+2(5a-3n+5)x^...
User8976's user avatar
  • 199
9 votes
2 answers
344 views

Counting $m\times n$ $\bigl({1\atop1}{1\atop0}\bigr)$-free $(0,1)$-matrices

Let $G_{m,n}$ denote the number of $m\times n$ $(0,1)$-matrices that avoid the submatrix $\bigl({1\atop1}{1\atop0}\bigr)$. (Submatrices need not be contiguous.) Here are some small values (not yet on ...
ho boon suan'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
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
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
6 votes
1 answer
288 views

The combinatorics of the Nullstellensatz for the variety of nilpotent matrices

Let $H_n$ denote the set of $n \times n$ nilpotent matrices with complex entries. The set $H_n$ may be regarded as an algebraic variety. Indeed, consider the polynomial ring $\mathbb{C}[A_{i,j} : 1 \...
Samuel Johnston's user avatar
2 votes
2 answers
217 views

Questions regarding answer to complex symmetric square root of a complex symmetric invertible matrix

I am looking at showing that a complex symmetric invertible matrix always has a complex symmetric square root and I refer to this Q&A for the answer to this question. I am little confused at the ...
jcb2535's user avatar
  • 57
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
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
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
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
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
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
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
2 votes
0 answers
233 views

Roots of determinant of matrix with polynomial entries — a generalization

For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
GA316's user avatar
  • 1,269
2 votes
1 answer
417 views

Roots of determinant of matrix with polynomial entries

Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying $$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
GA316's user avatar
  • 1,269
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
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
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
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
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
2 votes
1 answer
244 views

radical of a certain ideal of sixteen variable polynomial ring, generated by the entries of certain matrices

Consider the polynomial ring $R=\mathbb C[x_1,x_2,...,x_{16}]$, and set $$X=\begin{pmatrix} x_1 &x_2&x_3 &x_4\\ x_5&x_6& x_7&x_8\\x_9&x_{10}&x_{11}&x_{12}\\x_{13}&...
user521337's user avatar
  • 1,209
8 votes
1 answer
441 views

A question on symmetric matrices

$\newcommand{\R}{\mathbb{R}}$ The question is Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
Iosif Pinelis'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
6 votes
1 answer
277 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
Mahdi - Free Palestine's user avatar
8 votes
0 answers
176 views

Nonzero subdeterminants conjecture: has anybody seen this anywhere?

I already posted this question on Mathematics StackExchange. A user there suggested that I rather post it on mathoverflow, since it is a research question. So here it is. Let $m\geq2$, $n\geq1$ be ...
chizhek's user avatar
  • 291
2 votes
0 answers
208 views

Real-rooted polynomials with coefficient constraints

My question is whether there exists $(a_0, a_1, \ldots, a_{2n-1}) \in \mathbb{R}_{+}^{2n}$ such that (1). $a_{2k} + a_{2k+1} = \binom{3n-1}{3k} + \binom{3n-1}{3k+1} + \binom{3n-1}{3k+2}$ for all $0 \...
KDD's user avatar
  • 151
5 votes
1 answer
473 views

higher order analogues of sylvester's law of inertia?

Sylvester's law of inertia (here I quote wikipedia) If A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAS^{T} is diagonal, then the number ...
mathstudent42's user avatar
0 votes
1 answer
141 views

Searching for matrices with some property

I don't know if this question is considered research-related. If not, I will move it to Math SE. I am searching for matrices with the property $$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) ...
user avatar
1 vote
1 answer
925 views

What is special about 2 + $\sqrt{3}$?

Well, one thing is special about it, but it takes a while to explain. Please let me know, whether this number occurs in other special occasions as well. The explanation: Let $p$ be a complex ...
thomashennecke's user avatar
5 votes
0 answers
205 views

Resultant of a binomial and a trinomial

Does anyone know of any papers dealing with the resultant of a binomial $x^n+a$ and a trinomial $x^r+bx^s+c$ ? Even special cases would be of interest. (The resultant of two binomials is well known.)...
Gary McGuire's user avatar
3 votes
0 answers
182 views

Diagonalization of Hermitian Matrix Polynomials

I have a question on the decomposition of polynomial matrices. Suppose $A(\lambda) = \sum_{j=0}^L \lambda^j A_j$ is an $n \times n$ matrix of polynomials, which is Hermitian on the real axis $\lambda ...
William's user avatar
  • 31
4 votes
1 answer
277 views

Polynomials on spaces of matrices

Let $\mathbb{P}^N$ be the projective space parametrizing $n\times n$ non-zero matrices modulo scalar multiplication, and let $\mathbb{P}^M\subset\mathbb{P}^N$ be the subspaces of symmetric matrices. ...
user avatar
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
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
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
69 views

A possible conjecture on exponential asymptotics of random recursion relations

I have come to suspect that the following is true (and have confirmed it with some numerical experiments) but I have no idea how to prove it. Background: Let $f(z) = \sum_{n=0}^N a_n z^n$ be some ...
Ruben Verresen's user avatar
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
1 vote
1 answer
129 views

On Polynomial Characterization of Projection area of semidefinite matrices

Suppose $m,n$ are positive integers. $D$ denotes the set of $n\times n$ complex semidefinite positive matrices with unit trace. $A_1,\cdots,A_m$ are $n\times n$ Hermitians. We are interested in the ...
gondolf's user avatar
  • 1,503
-2 votes
1 answer
158 views

About local maxima of multivariable polynomials

Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
gradstudent's user avatar
  • 2,246
4 votes
3 answers
667 views

Regularity for the roots of (characteristic) polynomials with given multiplicity

A classical result states that roots of a polynomial are continuous functions of its coefficients. This is, for exemple, a direct consequence of Rouché's theorem. Using the implicit function ...
Adrien Hardy's user avatar
  • 2,135
2 votes
1 answer
254 views

Polynomials and matrices in $\Bbb F_q$

Given a polynomial $p(x,y)\in\Bbb F_q[x,y]$ of $(x,y)$ degree $(n_x,n_y)$ ($n_x,n_y\geq0$ and $n_x,n_y\in\Bbb Z$) where $q=p^\alpha$ with $p$ a prime and $\alpha\in\Bbb N$ how many different matrices $...
Turbo's user avatar
  • 13.9k
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
0 votes
1 answer
193 views

Find the roots of polynomial equation that is from a quadratic matrix

Assume the polynomial equation from a quadratic matrix is written as $$P=XHX^T$$ where $X=[1,x,x^2,\cdots,x^M ]$, $H$ is a symmetric matrix of order $(M+1)$. How do we find the roots of the above ...
Dings's user avatar
  • 153
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