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# Matrix and least squares

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Let P = A(A^TA)^-1A^T, where A is an mxn matrix of rank n.

(1) Show that P^2 = P.
(2) Prove P^k = P for K = 1,2,....
(3) Show that P is symmetric.

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Let A&#1028;R^(mxn) and let r be a solution to the least square problem Ax=b. Show that a vector y&#1028;R^n will also be a solution if and only if y = r + z, for some vector z&#1028;N(A).

##### Solution Summary

There are several proofs regarding matrices here, including a proof of a symmetric matrix.

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