Show that the collection of matrices which commute with every idempotent matrix are the scalar matrices

Question

Show that the collection of $n\times n$ real matrices which commute with every idempotent matrix are the scalar matrices .

Let $\mathcal P$ denote set of all idempotent matrices .

Let $A=\{B:BP=PB\forall P\in \mathcal P\}$.

So I need to show that $A=\{cI:c\in \Bbb R\}$.

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I am feeling totally confused on this. Will you kindly give some hints here?

Answer 1

Let $B$ be a matrix commuting with every idempotent matrix. Fix any vector $u$, I claim that $u$ and $Bu$ cannot be both nonzero and linearly independent.

If they were, then we could take $P$ to send $Bu$ to $u$, and $u$ to itself. But then $$ Bu = B(Pu) = P(Bu) = u, $$ contradicting that $u$ and $Bu$ were linearly independent.

This implies that $Bu = \lambda_u u$ for every $u \in \mathbb{R}^n$. I.e., every vector is an eigenvector of $B$.

Taking $u$ and $v$ to be any two linearly independent vectors, then $B(u + v) = \lambda_u u + \lambda_v v$ has to be a scalar multiple of $u + v$, and we conclude that $\lambda_u = \lambda_v$.

So there is a single scalar $\lambda$ such that $Bu = \lambda u$ for all $u$, which is what we wanted to show.

Answer 2

Eigen value of projection mapping is 1 since it only projects doesn't stretch Now see if Eigen value of scaler matrix is c of whole vector space V where c is diagonal element of scaler matrix

Now you can show only scaler matrix will have same Eigen vale by pre multiplication and post multiplication to idempotent matrix that is c Any other linear map upto similarity has Eigen value of different different subspaces of V