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Let H = X(X'X)^-1X' and M = In - X(X'X)^-1X' where X is a n x...

Let H = X(X'X)^-1X' and M = In - X(X'X)^-1X' where X is a n x k-dimensional matrix.

Show that H is positive semi-definite and has rank k. Use that a symmetric matrix is positive semi-definite if and only if its eigenvalues are non-negative. You can use that rank(H) = trace(H).

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