How to Prove a Matrix is Idempotent?

mattson17 · Sep 25, 2008

A=I - X(X'X)^-1X'

Show that Matrix A is idempotent.

I'm new to matrices and am having trouble proving this. Could anyone give me a hand as far as how to get started on solving this problem and possibly some tips for how to do problems like it. Thanks.

statdad · Sep 25, 2008

I'm guessing this question comes from statistics (multiple linear regression?), because the matrix

[tex] I - X(X'X)^{-1}X'[/tex]

generates the residuals in that topic.

Two ways to go - neither is any better than the other
1) Show the "X" portion itself is idempotent, then work with the entire thing
2) Move directly to working with the entire expression

Either approach requires judicious use of parentheses and the associative property of matrix multiplication.

How to Prove a Matrix is Idempotent?

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