Let the n n matrices A and B be similar Prove that A Ink a

Let the n × n matrices A and B be similar.

Prove that (A + (I_n))^k and (B + (I_n))^k are similar for all integers k 1 and R.

Solution

We know that a n × n matrix A is is similar to a n x n matrix B if B = P^-1 AP for some invertible n x n matrix P. We will first prove that A^k and B^k are similar matrices when k 1.

Since A and B are similar, there is some nonsingular matrix P such that B = P^-1 A P. For k = 1, apparently A and B are similar matrices. When k=2, we have B²=B.B = (P^-1AP)(P^-1AP) = P^-1A(PP^-1)AP = P^-1A( I_n )AP = P^-1AAP = P^-1A²P. Therefore, B² and A² are similar matrices. Thus A^k and B^k are similar matrices when k = 1 or 2. We can extend this argument to show that A^k and B^k are similar matrices when k = 3,4,… etc. Thus, by induction, A^k and B^k are similar matrices when k 1.

Now, P^-1I_nP = P^-1 P = I_n , so B + I_n = P^-1 AP + P^-1 I_n P = P^-1AP+ P^-1I_n P = P^-1 (AP+ I_n P) = P^-1(A+ I_n) P. Therefore, B + I_n is similar to A + I_n. Hence, from the previous part, (A +I_n)^k is similar to (B +I_n)^k.