This question is inspired by a sentence in Roe's "Elliptic Operators, Topology, and Asymptotic Methods." An invariant polynomial $P$ on the matrix lie algebra $\mathfrak{gl}_m(\mathbb{C})$ is defined to be a polynomial in the entries of the matrix such that $P(AB) = P(BA)$ for any $A,B \in \mathfrak{gl}_m(\mathbb{C})$.
Roe writes the following: "Let $P$ be any invariant polynomial. If we first of all look at the restriction of $P$ to diagonal matrices, we see that $P$ must be a polynomial function of the diagonal entries."
If we consider diagonal matrices, $AB = BA$, so $P(AB) = P(BA)$ for literally any polynomial $P$, invariant or not. I don't see how we can deduce anything about $P$ by restricting to diagonal matrices, much less that it only involves the diagonal entries of a matrix. Can anyone decipher what Roe means here?
This is not true, the determinant is an invariant polynomial, and $det\pmatrix{0&x\cr 1&0}=-x$. Perhaps you want to say that it depends of the eigenvalues if $A$ is a diagonalizable matrix since you can find an invertible matrix $B$ such that $BAB^{-1}$ is diagonal and $P(BAB^{-1})=P(A)$.