TITLE: NP-hard problems in linear algebra and real analysis
QUESTION [8 upvotes]: I am curious about NP-hard problems in linear algebra and real analysis. An example in linear algebra would be the calculation of the permanent.
I would thus like to collect in this thread a list of problems specifically in real analysis and linear algebra which are proven to be NP-hard. References to articles or books on this subject would also be helpful.
Any help will be greatly appreciated.

REPLY [9 votes]: The tensor product has a way of making easy problems into (NP-)hard problems. Rank of a 2-tensor (matrix)? Easy. Rank of a 3-tensor? NP-hard. Spectral norm of a matrix? Easy. Spectral norm of a 3-tensor? NP-hard. Etc. There is a nice article by Hillar and Lim about this.
As a direct corollary of the above fact, many natural linear algebra problems that come up in quantum information theory are NP-hard:

Determining whether or not a quantum state is entangled
Computing pretty much any entanglement monotone (squashed entanglement, entanglement cost, relative entropy of entanglement, and so on)
Holevo capacity of a quantum channel
Minimum entropy of a quantum channel
Plenty of other problems are even "harder" than NP-hard (see this survey article)

These problems can sometimes be rephrased in a way that doesn't use the tensor product at all even. For example, the problem of determining whether or not a linear map is positive is NP-hard. That is, given a linear map $\Phi$ that sends matrices to matrices, determine whether or not $\Phi(X)$ is positive semidefinite whenever $X$ is positive semidefinite.

REPLY [4 votes]: Given a symmetric matrix do determine it whether it is copositive is NP-Hard.