tracial state of a orthogonal projection

Question

Suppose $A\in M_n(\mathbb{C})$,$A$ has eigenvalues$\lambda_1,\cdots,\lambda_n$,$P$ is the orthogonal projection from $\mathbb{C}^n$ onto the span of eigenvectors associated with $\lambda_1,\cdots,\lambda_n$,how to computer the tracial state of $P$.I saw a reference book,it writes:$tr_n(P)=\frac{1}{n}$(cardinality of $\lambda_1,\cdots,\lambda_n)$.I feel a little confused.

Answer 1

There is no general answer to this. If $\lambda_1,\ldots,\lambda_n$ are distinct, then the eigenvectors of $A$ span $\mathbb C^n$ and so $P=I$. If $A=\lambda P$, then $\operatorname{tr}_n(P)=k/n$, where $k$ is the rank of $P$.

So all possible values $1/n,2/n,\ldots,1$ are possible.