1
vote
0answers
31 views

Is there any significance to this matrix/operator?

I am working on a problem involving the the polarized Hessian covariant in Cartesian coordinates on $\mathbb{R}^2$ $[a,b] = \frac{1}{2} \frac{\partial ^2 a}{\partial x ^2} \frac{\partial ^2 ...
1
vote
0answers
67 views

Invariants of the Determinant Form

Consider a form of degree $r$ in $n$, that is, a homogeneous polynomial $$f(x_1, \ldots, x_n)=\sum_{i_1+\ldots i_n=r}\alpha_{i_1 ... i_n}x_1^{i_1} ... x_n^{i_n} $$ After the linear change of ...
0
votes
0answers
96 views

“Invariant integral” for linearly reductive groups, and the Reynolds operator

Let $G$ be an affine algebraic group. Let $$\lambda\colon G\to GL(k[G]),\quad \lambda(g)(f)=(h\mapsto f(g^{-1}h)),$$ be the left-translation, which is a rational representation of $G$ on its ...
2
votes
0answers
93 views

Invariant polynomials over symmetric matrices under Euclidean transformations

It is a simple question, but I haven't still had a course on this topic and I'm finding it hard to understand some basics. Consider a $2\times2$ symmetric matrix over a field (for example ...
0
votes
1answer
195 views

Invariant space of linear transformation

Let $V$ be a vector space of a finite nonzero dimension $n$ over some field. Let $T$ be a linear transformation of $V$, such that $T$ is nonzero and not one-to one. (a)Give a $T$-invariant linear ...
3
votes
1answer
1k views

Invariant Factors vs. Elementary Divisors

I have been studying Cooperstein's Advanced Linear Algebra for about seven months now and I am having problems understanding how to find the elementary divisors of a linear operator and how to find ...
2
votes
0answers
115 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...