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

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) (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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Start with the given equation.

Right multiply both sides by C (see note below)

Plug in on the right side.

Use the associative property of matrix multiplication.

The expression is equal to since X is nonsingular (ie X is invertible)

Use the associative property of matrix multiplication.

Matrix Multiplicative Identity Property

Use the associative property of matrix multiplication.

Matrix A is idempotent, so A*A = A

Replace the right side with what it really is

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Because we've shown that 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.

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SOLUTION: Let A be an idempotent matrix and X a nonsingular matrix. Show that C= XAX^&#8722;1 is an idempotent matrix.

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