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Results tagged with matrix-theory
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user 13972
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
0
votes
Number of Skew Symmetric Matrices of fixed rank
Offhand, I don't know a reference, but I'm sure that the transitivity result can be found in some book that treats Lie algebras over finite fields. However, it's easy enough that you don't need to go …
- 108k
8
votes
Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces
$\DeclareMathOperator\GL{GL}$I think that there is probably too much variety in the examples to expect any clean classification. Consider the following examples:
First, as I mentioned in my comment, …
- 108k
15
votes
Parametrization of positive semidefinite matrices
To get a parameterization of the kind you want, the space $S_{n,r}$ of positive semidefinite symmetric $n$-by-$n$ matrices of rank $r$ (with ($0<r<n$) would have to be contractible, but it is not. I …
- 108k
10
votes
Almgren's regularity Theorem ; a simple example?
As I mentioned in my comment, a pair of orthogonal $2$-discs in $\mathbb{R}^4$ is known to be minimizing among all orientable $2$-currents with the same boundary by the technique of calibrations: One …
- 108k
28
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries.
One proof is using representation theory: If we let $S\subset M_ …
- 108k
7
votes
Accepted
Isomorphism between the hyperbolic space and the manifold of SPD matrices with constant dete...
This 'isomorphism' does not hold for $n>2$, in the sense that the 'natural' $\mathrm{SL}(n,\mathbb{R})$-invariant metric on what you are calling $\mathcal{SP}(n)$ and the constant sectional curvature …
- 108k