How can I project matrix $A$ onto the space of the matrix $X$?

Question

I don't know if this question makes any sense, but how can I project matrix $A$ onto the space of the matrix $X$ (how can I compute $\textrm{Projection}_{X}A$)?

Thank you,

Answer 1

The question makes sense: with $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, in the vector space $\mathcal{M}_n(\mathbb{K})$, you want to find the projection on $\mbox{Span}(X)$ (assuming that $X$ is not zero, otherwise it is obvious).

A good candidate is $P(M) = \frac{\mbox{Tr}(M^TX)}{\mbox{Tr}(X^TX)}X$. Then the image of $P$ is $\mbox{Span}(X)$ and for any $\lambda \in \mathbb{K}$, you find $P(\lambda X) = \lambda X$.