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Results tagged with matrix-analysis
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user 48839
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
3
votes
Accepted
Symmetric and anti-symmetric matrices and maximal eigenvalues
If $iA_1v=\eta v$, $\|v\|=1$, then $w=(|v_1|,\ldots ,|v_n|)$ is still normalized and $\langle w, Aw\rangle \ge \langle v, iA_1 v\rangle =\eta$, so the claim follows from min-max.
- 24.2k
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
4
votes
Inequality between nuclear norm and operator norm for positive definite matrices
There is essentially no $n$ dependence here. It suffices to consider the case $k=1$ (by rescaling), and then $c=1/2$ works for all $n$. This follows because
$$
\|B^{-1/2}AB^{-1/2} \| \ge 1 ,\quad\quad …
- 24.2k
1
vote
Maximizing trace of $\mathrm V^T \mathrm A \mathrm V$ for $\mathrm A$ symmetric (alternate p...
I think your simple argument is perfectly appropriate. We could bring in min-max as follows: as already pointed out by Nick and yourself, the only thing that's not immediately clear is the estimate
$$ …
- 24.2k
18
votes
Accepted
Determinant of a $k \times k$ block matrix
We can just manipulate $C$ in the usual way by row operations: Subtract the last "row" from all the other "rows" (this is really several traditional row operations done at once). This produces
$$
\beg …
- 24.2k
2
votes
Relation between the subordinate norm and the spectral radius of a matrix
This ratio $\|A\|_{2,b}/\rho(A)$ can be arbitrarily small. Consider the $N\times N$ matrix $A$ all of whose entries are equal to $1$, and interpret this as a block matrix with blocks of size $M=1$. Th …
- 24.2k