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kingwinner
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Q: Suppose the only eigenvalues of A are 1 and -1, and A is similar to a diagonal matrix. Prove that A^-1 = A
My Attempt:
Suppose the only eigenvalues of A are 1 and -1, and A is similar to a diagonal matrix.
=>A is invertible (since 0 is not an eigenvalue of A)
and there exists invertible P s.t.
(P^-1) A P = D is diagonal
=> A= P D (P^-1)
=> A^-1 = P (D^-1) (P^-1)
Now if I can prove that D = D^-1, then I am done. But I am stuck right here. The trouble is that the eignevalues of A can have any number of multiplicities, so D can be diag{1,1,-1,-1,-1}, diag{1,-1,-1,-1,-1,-1}, etc., there are infinite number of possible D's, how can I prove that D = D^-1 is always true for these infinite number of different settings?
Can someone please help me?
Thanks a lot!
My Attempt:
Suppose the only eigenvalues of A are 1 and -1, and A is similar to a diagonal matrix.
=>A is invertible (since 0 is not an eigenvalue of A)
and there exists invertible P s.t.
(P^-1) A P = D is diagonal
=> A= P D (P^-1)
=> A^-1 = P (D^-1) (P^-1)
Now if I can prove that D = D^-1, then I am done. But I am stuck right here. The trouble is that the eignevalues of A can have any number of multiplicities, so D can be diag{1,1,-1,-1,-1}, diag{1,-1,-1,-1,-1,-1}, etc., there are infinite number of possible D's, how can I prove that D = D^-1 is always true for these infinite number of different settings?
Can someone please help me?
Thanks a lot!