How to Prove a Matrix is Idempotent?

In summary, the conversation discusses how to show that Matrix A, represented by I - X(X'X)^-1X', is idempotent. The person asking for help is new to matrices and is looking for tips on how to solve this type of problem. The conversation suggests two ways to approach the problem, both of which involve using parentheses and the associative property of matrix multiplication.
  • #1
mattson17
1
0
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.
 
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  • #2
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.
 

1. What is an idempotent matrix?

An idempotent matrix is a square matrix that, when multiplied by itself, results in the same matrix. In other words, the matrix remains unchanged after repeated multiplication.

2. How do you prove that a matrix is idempotent?

To prove that a matrix is idempotent, you can simply multiply the matrix by itself and see if the result is equal to the original matrix. This can also be proven algebraically by using the definition of an idempotent matrix.

3. What are the properties of an idempotent matrix?

Some properties of an idempotent matrix include having eigenvalues of either 0 or 1, having a determinant of 0 or 1, and being both diagonalizable and symmetric.

4. Can a non-square matrix be idempotent?

No, a non-square matrix cannot be idempotent because the definition of an idempotent matrix requires the number of rows and columns to be equal.

5. How is an idempotent matrix used in real life?

Idempotent matrices have various applications in fields such as economics, statistics, and computer science. Some examples include using idempotent matrices in linear regression models, Markov processes, and data compression algorithms.

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