Due on Friday, Feb 15.
Prove that similarity is an equivalence relationship.
Suppose that \(A, B \in M_n\) and \(A\) is similar to \(B\). For any polynomial \(p(t)\), prove that \(p(A)\) is similar to \(p(B)\).
Suppose \(A \in M_n\) is diagonalizable, and let \(p_A(t)\) denote the characteristic polynomial of \(A\). Prove that \(p_A(A)\) is the zero matrix. Hint: Prove for any polynomial \(p\) and diagonal matrix \(D =\operatorname{diag}(d_1,d_2,\ldots,d_n)\), that \(p(D) = \operatorname{diag}(p(d_1),p(d_2),\ldots, p(d_n))\).
Recall that the rank of a matrix is the dimension of its range (which is the same as the dimension of the column space). Prove that rank is a similarity invariant, that is, two similar matrices must have the same rank. Hint: Use the fact that if \(S\) is an invertible matrix, then \(\operatorname{rank}(SA) = \operatorname{rank}(A)\), and also \(\operatorname{rank}(BS) = \operatorname{rank}(B)\).
Show that the rank of a diagonalizable matrix is the number of non-zero eigenvalues.
Give an example of a non-diagonalizable matrix with a rank that is not the same as the number of non-zero eigenvalues.