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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.
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
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 …
- 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} …
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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 …
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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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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
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 …
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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 …
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