Question

Suppose P is invertible and \(A = PB{P^{ - 1}}\). Solve for Bin terms of A.

Short answer

\(B = {P^{ - 1}}AP\)

Step-by-step solution

Condition for an invertible matrix

Theorem 5states that Ais an invertible \(n \times n\) matrix, then for each b in \({\mathbb{R}^n}\), the equation \(Ax = b\) has a unique solution \(x = {A^{ - 1}}b\).

Solve for B in terms of A

Multiply both sides of the equation \(A = PB{P^{ - 1}}\) by \({P^{ - 1}}\):

\(\begin{aligned}{c}{P^{ - 1}}A = {P^{ - 1}}PB{P^{ - 1}}\\{P^{ - 1}}A = IB{P^{ - 1}}\\{P^{ - 1}}A = B{P^{ - 1}}\end{aligned}\)

Multiply both sides of theobtainedequation by P:

\(\begin{aligned}{c}{P^{ - 1}}AP = B{P^{ - 1}}P\\{P^{ - 1}}AP = BI\\{P^{ - 1}}AP = B\end{aligned}\)

Thus, \(B = {P^{ - 1}}AP\).