How to find a matrix closest to a given matrix in a Inner product space?

Question

Consider $M_2(\mathbb{C})$ with the inner product $$\langle A, B \rangle = trace(B^*A)$$ where $*$ is conjugate transpose. Find the closest element of the complex symmetric $2\times 2$ matrices to $$A = \begin{bmatrix}1&-i\\ i&1\end{bmatrix}$$How to approach this problem?

Answer 1

1. Pick a basis $(e_1,e_2,e_3)$ of the space of all $2\times2$ matrices.
2. Use Gram-Schmidt to create from it an orthnormal basis $(f_1,f_2,f_3)$.
3. The answer to your problem will be $\langle A,f_1\rangle f_1+\langle A,f_2\rangle f_2+\langle A,f_3\rangle f_3$