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Please help me solve this problem:

Find matrix $A$ such that $A$ commutes with all projection matrix (A matrix $P$ is said to be projection if $P^2 = P$).

Thank you so much.

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    Hint: such a matrix has to be diagonalizable in every basis, since it leaves every one-dimensional subspace invariant (as it commutes with the projection onto the latter). – Julien Nov 18 '13 at 14:07
  • I'm sorry, I should make it more clear. The problem requires us find ALL matrices $A$.It is not difficult to prove that $A$ must be a diagonal matrix by consider all matrices $P$ with only $i$-th ($i = 1, 2, \dots, n$) element in the diagonal being 1, all others being 0.

    How to do next, could we prove that all diagonal matrices are solution or the matrices must have some other properties?

     –  Nov 18 '13 at 14:16
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    I understood the question, it is clear enough. My hint is supposed to let you conclude more than merely "diagonal". Basically you should note that $Ax=\lambda_x x$ for every vector $x$. Then prove that the scalar $\lambda_x=\lambda$ does not depend on $x$. – Julien Nov 18 '13 at 14:21
  • @julien IMO, this should be an answer. – Vedran Šego Nov 18 '13 at 14:44
  • Excellent idea :). Thank you so much. –  Nov 18 '13 at 14:55

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