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IMC 1994 D2 P4

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March 6, 2017
IMClinear algebramatrix

Problem Statement

Let AA be a n×nn\times n diagonal matrix with characteristic polynomial (x−c1)d1(x−c2)d2…(x−ck)dk(x-c_1)^{d_1}(x-c_2)^{d_2}\ldots (x-c_k)^{d_k} where c1,c2,…,ckc_1, c_2, \ldots, c_k are distinct (which means that c1c_1 appears d1d_1 times on the diagonal, c2c_2 appears d2d_2 times on the diagonal, etc. and d1+d2+…+dk=nd_1+d_2+\ldots + d_k=n).
Let VV be the space of all n×nn\times n matrices BB such that AB=BAAB=BA. Prove that the dimension of VV is d12+d22+⋯+dk2d_1^2+d_2^2+\cdots + d_k^2
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