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Questions about the properties of vector spaces and linear transformations, including linear systems in general.

34 votes
8 answers
4k views

Uncountable counterexamples in algebra

In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a separable …
21 votes
6 answers
2k views

What are the possible eigenvalues of these matrices?

Edit: since we seem a bit deadlocked at this point, let me weaken the question. It's fairly easy to see that the set of 8-tuples of reals which can be the eigenvalues of a matrix of the desired form i …
12 votes
2 answers
1k views

Another $2 \times 2$ matrix question

This question is similar to this previous one but I think it is harder. Let $X$, $Y$, $Z$, and $W$ be $2\times 2$ Hermitian matrices. Can we always find $\theta,\phi \in [0,\pi/2]$ and $2\times 2$ un …
15 votes
1 answer
1k views

Existence of double eigenvalue

Let $A$ and $B$ be complex $4\times 4$ matrices. Assume both are Hermitian, and that they are linearly independent. Must there exist a nonzero real linear combination $aA + bB$ which has a repeated e …
20 votes
3 answers
1k views

Simultaneous "orthonormalization" in $\mathbb{C}^4$

Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix. So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good fo …
10 votes
1 answer
618 views

$2 \times 2$ matrix question

Let $A$, $B$, and $C$ be $2\times 2$ complex matrices, with $A$ and $C$ rank $1$ Hermitian. Can we find a real number $a$ and a $2\times 2$ unitary $U$ such that $$A + BV + V^*B^* + V^*CV$$ is a scala …
7 votes
1 answer
409 views

Geometry of Hermitian rank $\leq r$ matrices

Let $M_n^{sa}$ be the space of $n\times n$ complex Hermitian matrices, let $r < n$, and let $E$ and $F$ be (real) linear subspaces of $M_n$ with ${\rm codim}(E) < r^2$ and ${\rm codim}(F) = 1$. Let $V …
52 votes
2 answers
3k views

vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct. This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest po …