SOLUTION: Let A be an idempotent matrix and X a nonsingular matrix. Show that C= XAX^−1 is an idempotent matrix.

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Question 904029: Let A be an idempotent matrix and X a nonsingular matrix. Show that C= XAX^−1 is an idempotent matrix.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
We want to show that C is an idempotent matrix

http://mathworld.wolfram.com/IdempotentMatrix.html

That means we want to show that C*C = C is true.

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C+=+X%2AA%2AX%5E%28-1%29 Start with the given equation.

C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2AC Right multiply both sides by C (see note below)

C%2AC+=+%28X%2AA%2AX%5E%28-1%29%29%2A%28X%2AA%2AX%5E%28-1%29%29 Plug in C+=+X%2AA%2AX%5E%28-1%29 on the right side.

C%2AC+=+X%2AA%2A%28X%5E%28-1%29%2AX%29%2AA%2AX%5E%28-1%29 Use the associative property of matrix multiplication.

C%2AC+=+X%2AA%2A%28I%29%2AA%2AX%5E%28-1%29 The expression X%5E%28-1%29%2AX is equal to I since X is nonsingular (ie X is invertible)

C%2AC+=+X%2A%28A%2AI%29%2AA%2AX%5E%28-1%29 Use the associative property of matrix multiplication.

C%2AC+=+X%2AA%2AA%2AX%5E%28-1%29 Matrix Multiplicative Identity Property

C%2AC+=+X%2A%28A%2AA%29%2AX%5E%28-1%29 Use the associative property of matrix multiplication.

C%2AC+=+X%2AA%2AX%5E%28-1%29 Matrix A is idempotent, so A*A = A

C%2AC+=+C Replace the right side with what it really is

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Because we've shown that C%2AC+=+C is true, this proves that C is an idempotent matrix.


Note: You can also use left multiplication to get to the same result (in a very slightly different way, nothing too major though). Left multiplication is a bit different than right multiplication because matrix multiplication is NOT commutative. The proof is effectively the same as shown above which is why I don't need to show it, but it's important to keep this in mind.