Ytukyg

Let $A$ be a $4 times 4$ complex diagonal matrix with exactly three distint entries on its diagonal.

(1) What is the dimension of the vector space of polynomials of $A$?

(2) What is the dimension of the vector space of $4 times 4$ complex matrices that commute with $A$?

(3) If $B$ is a $4 times 4$ complex diagonal matrix with exactly three distinct entries on its diagonal, is it similar to a polynomial of $A$?

I am looking for explainations more so than the actual answers.

In order to make writing things easier, I will specifically consider the case where the repeated eigenvalue comes first. That is, we have
$$
A = pmatrix{lambda_1 \ & lambda_1 \ && lambda_2 \ &&& lambda_3} = pmatrix{lambda_1 I_{2}\ & lambda_2 \ && lambda_3}
$$
where $I_2$ denotes a size $2$ identity matrix.

Hint for 1: Note that for any polynomial $p$,
$$
p(A) = pmatrix{p(lambda_1)I_{2} \ & p(lambda_2) \ && p(lambda_3)}
$$

Hint for 2: Verify that any block matrix of the form
$$
B = pmatrix{B_1\ & b_2 \ && b_3}
$$
will commute with $A$, where $B_1$ can be any $2 times 2$ matrix and the $b_i$ are scalars. Note too that these are the only matrices that commute with $A$.

Hint for 3: Every polynomial of $A$ is diagonal (and therefore diagonalizable). However, the matrix
$$
pmatrix{lambda_1&1 \ & lambda_1 \ && lambda_2 \ &&& lambda_3}
$$
is neither diagonal nor diagonalizable.