Kunze Linear Algebra · Problem 6.2.14

Elementary Canonical Forms

Mathematics — Methods · commuting-matrices-with-diagonal

Problem

Let AA be an n×nn \times n diagonal matrix with characteristic polynomial (x−c1)d1⋯(x−ck)dk,(x - c_1)^{d_1} \cdots (x - c_k)^{d_k}, where c1,…,ckc_1, \ldots, c_k are distinct. Let VV be the space of n×nn \times n matrices BB such that AB=BAAB = BA. Prove that the dimension of VV is d12+⋯+dk2d_1^2 + \cdots + d_k^2.

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