How to prove that an orthogonal projection matrix is idempotent

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An orthogonal projection matrix is defined as a matrix that projects vectors onto a subspace while preserving the length of the projection. To prove that such a matrix P is idempotent, one must show that P^2 = P. The discussion highlights the equation A(A^T*A)^-1 A^T and seeks clarification on the relationship between matrix A and matrix P. Understanding the definition of an orthogonal projection matrix is crucial for the proof. The conversation emphasizes the need for a solid grasp of the underlying concepts to successfully demonstrate the idempotency of the projection matrix.
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Homework Statement


Prove that [P]^2=[P] (that the matrix is idempotent)


Homework Equations





The Attempt at a Solution



A(A^T*A)^-1 A^T= (A(A^T*A)^-1 A^T)^2

Where A^T is the transpose of A. I have no idea.
 
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What is the relationship between your matrix A and your matrix P?

Backing up a bit further, what is the definition of an orthogonal projection matrix?
 

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