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1. Postdoctoral researcher in quantum informatics (concerning efficient algorithms to describe quantum mechanical systems, and/or how to use quantum mechanical systems to implement efficient algorithms).

2. Lover of toy problems and charicatures of The Big Picture alike, with a special interest in combinatorics, algebra, and phase transitions between them.

3. Lifelong enthusiast of mathematics.

Sep
24
awarded  Autobiographer
Jul
2
awarded  Curious
Apr
15
awarded  Popular Question
Feb
2
awarded  Yearling
Oct
17
accepted Condition number of a random 0-1 matrix
Oct
17
comment Condition number of a random 0-1 matrix
Because it's clear that this answer is still morally correct --- the 01 case is after all related to the symmetric case by a displacement by the all-1s matrix, which is very nearly orthogonal to all singular vectors but those of the largest singular value --- I'm accepting this answer. Perhaps these results also pertain exactly to the 01 case, though it's a bit frustrating that my certainty about this is undermined by a tacit but undeniable assumption of symmetry elsewhere in this work, which prevents me from being certain of the precise probabilistic lower bounds.
Oct
15
comment Condition number of a random 0-1 matrix
Unfortunately, it is difficult to tell whether or not the results in Lecture 18 of Vershynin's notes apply. While these would be the sort of results that I want, a casual glance at the notes of Lecture 7 imply that he might be implicitly restricting to subgaussian distributions with mean 0. My Bernoulli variables have mean $\frac12$, of course, and while I can see how to translate the results for the largest singular value, I do not know how to perform this translation for the minimum singular value. Can you point me to results which explicitly accomodate distributions with non-zero mean?
Oct
15
comment Condition number of a random 0-1 matrix
This seems like an interesting tangent, but I don't detect the relation to the Bernoulli variables which I consider. I suppose asymptotically the matrix $A^{\mathsf T} A$ will look as though it has Gaussian coefficients, but as I think you're hinting (to a deleted comment?) in a comment to the original question, these are not independent.
Oct
14
asked Condition number of a random 0-1 matrix
Feb
2
awarded  Yearling
Feb
1
asked Presentation of tree decompositions (and related concepts) in terms of continuous maps?
Nov
7
revised Matrices that are > 1 in a sense
Replaced all references with the 2-norm by Frobenius norm.
Nov
7
comment Matrices that are > 1 in a sense
@Suvrit: good point. I'm used to dealing almost exclusively with normal operators, so the fact that they are not in fact equivalent in general had not occurred to me. I'll edit the answer.
Oct
26
awarded  Enthusiast
Oct
17
revised Matrices that are > 1 in a sense
Fixed typo
Oct
17
revised Matrices that are > 1 in a sense
Revised to answer the question for right-multiplication, and to work more directly with singular vectors
Oct
16
answered Matrices that are > 1 in a sense
Oct
11
revised How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
fixed typo, revised definitions of variables
Oct
11
revised How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?
added 191 characters in body
Oct
11
asked How does the Schmidt decomposition generalize to tensor products of several finite-dimensional systems?