Finding the Orthonormal Basis of a given Subspace

Question

Hi, So I am not sure how to do this problem.

I think what I have to do is find a basis for the space of 2x2 matrices that are orthogonal to the identity matrix s.t. the HS inner product. And from there find an orthonormal basis. But I am not sure how to find the initial basis with the given information?

Thanks

Answer 1

To find the initial basis simple note that $$ V = \{ A\in M_{2\times 2}(\mathbb{R}) \colon \mathrm{tr} (A) = 0 \}. $$

So your space has dimension 3, and a basis is formed by $$ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}. $$

However this is not orthogonal, but from here can you proceed?