Are there open problems in Linear Algebra?

Question

I'm reading some stuff about algebraic K-theory, which can be regarded as a "generalization" of linear algebra, because we want to use the same tools like in linear algebra in module theory.

There are a lot of open problems and conjectures in K-theory, which are "sometimes" inspired by linear algebra.

So I just want to know:

What are open problems in "pure" linear algebra? (Pure means not numerical!)

Thanks

Answer 1

What are open problems in "pure" linear algebra? (Pure means not numerical!)

Here is a list of problems in "pure" matrix theory/linear algebra:

1. The Hadamard conjecture, which asserts that a Hadamard matrix of order $4k$ exists for every positive integer $k$. Most matrix theorists regard this as the most important open problem in matrix theory.
2. If you ask Charlie Johnson (and I have), the most important open problem in matrix theory is the nonnegative inverse eigenvalue problem (see the survey written by Johnson, Marijuan, Paparella, and Pisonero: https://link.springer.com/chapter/10.1007/978-3-319-72449-2_10), which is to characterize the spectra of entrywise nonnegative matrices. More specifically, given a multiset of complex numbers $\Lambda = \{\lambda_1,\ldots,\lambda_n\}$, find necessary and sufficient conditions such that $\Lambda$ is the spectrum of an entrywise nonnegative matrix $A$.
3. There are several matrix-theoretic formulations of the Riemann hypothesis, most notably that involving random matrices and the Redheffer matrix.
4. The Jacobian conjecture (#16 on Smale's list or problems).
5. Crouzeix's conjecture (the most recent of the conjectures listed).