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Suppose P is invertible and A=PBP^(-1). Solve for B in terms of A.?

December 1, 2022 0

Multiply both sides on the left by P^(-1):

P^(-1)A = P^(-1)(PBP^(-1))

==> P^(-1)A = (P^(-1) P)BP^(-1)

==> P^(-1)A = I BP^(-1)

==> P^(-1)A = BP^(-1).

Next, multiply both sides on the right by P:

(P^(-1)A) P = (BP^(-1)) P

==> P^(-1)AP = B (P^(-1) P)

==> P^(-1)AP = BI

==> P^(-1)AP = B.

I hope this helps!
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First right multiply by P: AP = PBP^(-1)P = PB Now left multiply by P^(-1): P^(-1)AP = P^(-1)PB = B.