16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 6.2.14

Elementary Canonical Forms

Mathematics — Linear Algebra · commuting-matrices-diagonal

Problem

Let

A

be an

n \times n

diagonal matrix with characteristic polynomial

(x - c_1)^{d_1} \cdots (x - c_k)^{d_k},

where

c_1, \ldots, c_k

are distinct. Let

V

be the space of

n \times n

matrices

B

such that

AB = BA

. Prove that the dimension of

V

d_1^2 + \cdots + d_k^2