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Results tagged with matrix-analysis
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user 10265
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.
4
votes
Accepted
Operator norm of a masked SDP matrix
No, no such constant exists. For example, if $I = \{(i,j) \mid i<j\}$, then $\Sigma\mapsto \Sigma_I$ is the usual triangular projection, and the norm is of order $\log n$, see for example Norm of the …
- 9,486
3
votes
Accepted
trace norm of AGB, where G is Gaussian random matrix
[Edit: Now I answer all questions.]
The answer to the first question is yes, the answer to the second question is no, and the answer to the third question is if and only if $p \geq 2$ (only a guess i …
- 9,486
5
votes
Accepted
Norm of triangular truncation operator on rank deficient matrices
The ratio is of order $O(\ln r)$. This follows from the fact that the triangular truncation is bounded on the Schatten class $S^p$ (=the operators $A$ on $\ell^2$ such that $\|A\|_p:= (Tr (A^*A)^{p/2} …
- 9,486
3
votes
Accepted
Restricting a continuous positive-semidefinite function to a finite subset
The answer for countably infinite has been given by fedja and Uri Bader in the comments and is yes: put $f$ to $0$ outside of the subgroup generated by $E$, and leave $f$ unchanged on this subgroup.
…
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9
votes
Accepted
When does $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Ver...
The term "operator Lipschitz function" is definitely not reserved to the Hilbert-Schmidt norm. On the opposite, I would say that it is mostly used for the operator norm (but not only, see for example …
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3
votes
Uniform smoothness inequality for Schatten norms
According to the Pisier-Xu survey "Non-commutative $L^p$ spaces" https://www.zbmath.org/?q=an%3A1046.46048, this is proved in
Ball, Keith; Carlen, Eric A.; Lieb, Elliott H.
Sharp uniform convexity and …
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3
votes
Distance of low-rank matrices to the identity for the $\infty$-norm
Your simple lower bound is not so bad, in particular when $m$ is of order $cn$ for $0<c<1$.
Indeed, it follows from the answers to this question that there are unit vectors $u_1,...,u_n$ in $\mathbf{R …
- 9,486
14
votes
Accepted
Almost commuting unitary matrices
Edit Now this answers the first question for the operator norm and the normalized Hilbert-Schmidt norm.
The answer depends on the norm you are considering. The answer is no for the operator norm, but …
- 9,486