Kunze Linear Algebra · Problem 3.4.12

Linear Transformations

Mathematics — Methods · nilpotent-operator-normal-form

Problem

Let VV be an nn-dimensional vector space over the field FF, and let B={α1,…,αn}\mathfrak{B} = \{\alpha_1, \ldots, \alpha_n\} be an ordered basis for VV. (a) According to Theorem 1, there is a unique linear operator TT on VV such that Tαj=αj+1,j=1,…,n−1,Tαn=0.T\alpha_j = \alpha_{j+1}, \quad j = 1, \ldots, n-1, \qquad T\alpha_n = 0. What is the matrix AA of TT in the ordered basis B\mathfrak{B}? (b) Prove that Tn=0T^n = 0 but Tn−1≠0T^{n-1} \neq 0. (c) Let SS be any linear operator on VV such that Sn=0S^n = 0 but Sn−1≠0S^{n-1} \neq 0. Prove that there is an ordered basis B′\mathfrak{B}' for VV such that the matrix of SS in the ordered basis B′\mathfrak{B}' is the matrix AA of part (a). (d) Prove that if MM and NN are n×nn \times n matrices over FF such that Mn=Nn=0M^n = N^n = 0 but Mn−1≠0≠Nn−1M^{n-1} \neq 0 \neq N^{n-1}, then MM and NN are similar.

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