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Results tagged with matrix-theory
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user 48839
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.
4
votes
Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for ...
This is my comment above slightly expanded. Let's focus on $2\times 2$ matrices for convenience and let $A(z)$ be entire with $\det A=1$ (divide through by a holomorphic square root of $\det A$ if thi …
- 24.2k
3
votes
How expressive is $e^A$ in the sense of universal approximation?
Edit 2 (in fact a complete rewrite): The condition from Noam's comment is almost, but not quite, the right condition. That lies somewhere between Noam's condition and condition (C) below, but lies str …
- 24.2k
11
votes
Accepted
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathb...
We can do this by a calculation. The assumptions on the determinant and trace are equivalent to having eigenvalues $\lambda,1/\lambda$, with $\lambda>1$. We can rotate the first eigenvector to the $e_ …
- 24.2k
1
vote
Accepted
Redistribute diagonal entries of a matrix
Yes, this works. Or, to be more honest, I'm fairly confident it does, but I'm only going to give a sketch.
The basic step is: a given symmetric $2\times 2$ matrix $A$ is unitarily equivalent to one w …
- 24.2k
12
votes
Accepted
Finding the nearest matrix with real eigenvalues
I have no idea what is going on, but your conjecture is not correct. This is more transparent perhaps in the complex version. Consider
$$
A = \begin{pmatrix} i & a \\ 0&-i \end{pmatrix} , \quad a>0.
…
- 24.2k