What is the fastest way to find if a real nonsquare matrix is injective?

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For a very tall matrix (m \times n where m \gg n), an even faster thing should be to compute the condition number via:

sqrt(cond(Hermitian(A'A)))

since that reduces it to a small n \times n matrix (by a fast matrix multiplication) before taking the SVD (and the SVD for a Hermitian matrix is even faster, just a Hermitian eigenproblem).

Computing A'A loses you some accuracy in the ill-conditioned case, but that probably doesn’t matter if you are only using this to check whether the condition number is large.

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