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Results tagged with matrix-analysis
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user 766
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.
9
votes
Accepted
A question about the paper "The Condition Number of a Randomly Perturbed Matrix"
One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough.
Thanks for pointing out this ty …
- 114k
11
votes
Accepted
Product $PVPVP$ is elementwise nonnegative?
First, we repeat the arguments from this stackexchange answer. $P^{-1}$ is an $M$-matrix, and can thus be written as $s(I-A)$ for some positive $s$ and some $A$ with non-negative entries. As $P^{-1} …
- 114k
29
votes
Accepted
A curious determinantal inequality
Let $C := A^{1/2} (A+B) A^{1/2} + B^{1/2} (A+B) B^{1/2}$; this is a positive semi-definite matrix with the same trace as $(A+B)^2$. We show that the eigenvalues of $C$ are majorised by the eigenvalue …
- 114k
9
votes
A curious determinantal inequality
EDIT: the argument below is not correct, but I am leaving it here in case it is of use in locating a better solution.
By a limiting argument we may assume that $C := A+B$ is invertible. If we write
…
- 114k
5
votes
Accepted
Is the p-norm of a matrix strictly log-convex?
The answer is "no".
First, a trivial counterexample: let $A_n$ be the $n \times n$ matrix with all $1$s on the first row and zeroes elsewhere, then $\frac{1}{p} \mapsto \|A_n\|_p = n^{1/p}$ is log-co …
- 114k