All Questions
8 questions
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
9
votes
3
answers
589
views
Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a ...
- 7,300
2
votes
1
answer
341
views
Symmetric orthogonal matrices with constant diagonal entries
$\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\varepsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\...
- 128k
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 ...
- 128k
2
votes
1
answer
217
views
Diagonalising a symmetric matrix with polynomial entries
Suppose I have a symmetric $2$ by $2$ matrix $M$ whose $(i,j)$-th entry $F_{i,j}(\mathbf{x})$ belongs to $\mathbb{R}[x_1, \ldots, x_n]$ for each $i,j$. I know that for each $\mathbf{x} \in \mathbb{R}^...
- 3,625
2
votes
1
answer
460
views
dimensions of strata of Pfaffian varieties
Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
- 3,779
7
votes
2
answers
1k
views
Dimension of the nilpotent centralizer of a nilpotent matrix
Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the $n\...
- 768
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,795