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I have a question. I consider following matrix:

A = {{1, 0}, {0, Cos[b]}}

and i would like to find its nullspace. If b=Pi/2, rank of A is 1 and then I can calculate its nullspace. However, if I write matrix A in the above-mentioned symbolic form, Mathematica returns that nullspace does not exist. How can I solve this problem?

How can I ask to Mathematica, for what values of b nullspace of A is not empty?

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Yeah, symbolic computations in Mathematica are usually performed for generic choices of values for the symbols. But Eigensystem can help here:

{eigenvalues,eigenvectors} = Eigensystem[A]

{{1, Cos[b]}, {{1, 0}, {0, 1}}}

Now you can see: If Cos[b] == 0 then {0, 1} lies in the null space of A.

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  • $\begingroup$ Uha, thanks @MichaelE2! $\endgroup$
    – Henrik Schumacher
    Commented Apr 21, 2018 at 13:33
  • $\begingroup$ Is it possible to ask to mathematica for what values of b the rank of matrix A is equal to 1? $\endgroup$
    – Gae P
    Commented Apr 24, 2018 at 9:06
  • $\begingroup$ The matrix A has rank 1 if precisely one element of eigenvalues is nonzero. In the case of a 2 by 2 matrix, Mathematica can be asked like this: Reduce[Or @@ Thread[eigenvalues == 0], b] $\endgroup$
    – Henrik Schumacher
    Commented Apr 24, 2018 at 10:23

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