All Questions
10 questions
4
votes
0
answers
168
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 ...
- 188k
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
$$A = \left(\begin{array}{}
1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
- 3,691
3
votes
0
answers
295
views
Decomposition of a determinant
Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$.
Does there exist a ...
CommunityBot
- 1
2
votes
0
answers
233
views
Roots of determinant of matrix with polynomial entries — a generalization
For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of ...
- 1,269
2
votes
1
answer
417
views
Roots of determinant of matrix with polynomial entries
Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying
$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x)...
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 $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 109k
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 ...
- 4,713
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
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 $\...
- 109k
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 $A$ ...
- 66.3k