Lineer Algebra. Finding P^(-1) and P

Question

Show that A=[-19,26,-17,23] and B=[2,1,-1,2]  are similar matrices by finding an invertible matrix P satisfying A=P^(-1)BP.

P^(-1)=? P=?

Answer 1

apply the formula for diagonalizing matrices to A and B:

1) A = P_AD_AP_A^-1 and 2) B = P_BD_BP_B^-1

solving for each gives:

D_A = D_B = D = [2+i,0,0,2-i] which suggests A and B are similar, we prove by answering the problem substituting the common matrix D into the above formulas:

1) A = P_ADP_A^-1 and 2) B = P_BDP_B^-1

from 2) we have D = P_B^-1BP_B substituting into 1) we have:

A = P_AP_B^-1BP_BP_A^-1 or

A = P^-1BP where P^-1= P_AP_B^-1 and P = P_BP_A^-1

^{(note P is defined here using the convention for the definition of similar matrices, not diagonalization)}

the diagonalization process above gave us:

P_A = [21-i,21+i,17,17] and P_A^-1 = 1/34[17i, 1-21i, -17i, 1+21i]

P_B = [-i, i, 1, 1] and P_B^-1 = 1/2[i, 1, -i, 1]

thus computing:

P^-1 = P_AP_B^-1 = [1, 21, 0, 17]

P = P_BP_A^-1 = [1, -21/17, 0, 1/17]

the reader can confirm that P and P^-1 as calculated are inverses and satisfy A = P^-1BP.