Proving a Schur product matrix is nonsingular

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monguss
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Need help to prove (or disprove, hope not) this result:

Let Q=[q_ij] be a orthogonal matrix (Q^T*Q=Identity) and let Q2 be the matrix of the squares of the entries of Q, that is Q2=[q^2_ij]. I need to prove that Q2 is nonsingular.


Been trying with some results about the Hadamard or Schur porduct in the sense that Q2=Q.Q
where the dot . represents the Schur product!

Any ideas

Thanks

Monguss
 

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monguss said:
Need help to prove (or disprove, hope not) this result:

If you don't know what the answer is supposed to be, it helps to make some examples up. Write down three orthogonal matrices, square their entries and see if the results are all invertible. If not you're done, if yes maybe you'll get some insight into what exactly is making them invertible when examining the inverses