Let T be an n n matrix with some Jordan canonical form J. Recall that the
Fantastic news! We've Found the answer you've been seeking!
Question:
Let T be an n × n matrix with some Jordan canonical form J. Recall that the sizes of the Jordan blocks of J corresponding to an eigenvalue λ of T are completely determined by the sequence of integers dim(ker(T − λI) n).
(a) Prove that if T, S are similar (i.e., there is an invertible matrix U such that S = UT U−1), then for all n ∈ N, dim(ker(T − λI)n) = dim(ker(S − λI)n).
(b) Conclude that if S, T are similar, then they have the same Jordan canonical form J.
(c) Recall that rank(A) = rank(At) for any matrix A. Use this and the rank-nullity theorem to prove that T and Tt have the same Jordan canonical form.
Related Book For
Posted Date: