1.) Given two matrices, A and B, show that
A = [[1,1],[-1,4]], and
B =[[2,1],[1,3]] are similar.
Then, show that the matrix
C = [[3,1],[-6,-2]] and
D = [[-1,2],[1,0]] are NOT similar.
Could you check to see if I am right? The theorem for similar matrices states that if n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).
Or: Two n × n matrices A and B are similar, if there exists a non-singular n × n matrix P such that B = P−1AP.
(1*4) - (-1*1) = 5, so it is non-singular.
(2*3) - (1*1) = 5, so it is non-singular
(A - LI)x = 0
A = [[1-L, 1],[-1,4-L]]
(1-L)*(4-L) - (-1) = 0
Solving for L, we get the eigenvalues are 5/2+1/2*sqrt(5), 5/2-1/2*sqrt(5)
B = [[2-L, 1],[1, 3-L]]
(2-L)*(3-L) - 1*1 = 0
Solving for L, we get the eigenvalues are the same as above.
Is this sufficient to show that they are similar, since they both have the same eigenvalues?
To show C and D are not similar:
(3-L)*(-2-L) - (-6) = 0
Solving for L, we get the eigenvalues are 0 and 1.
(-1-L)*(0-L) - 2 = 0
Solving for L, we get the eigenvalues are 1 and -2.
Is this sufficient to prove that they are not similar, since they don't have the same eigenvalues?