(Eigenvalue Decomposition) In econometrics, another popular decomposition for symmetric matrices is the eigenvalue decomposition. It is always possible to find an orthogonal matrix R and a diagonal matrix A such that $$\Omega = RAR'$$. The columns of R are called eigenvectors and the diagonal elements of A are called eigenvalues. See also Theorem C.16 (Eigenvalue Decomposition, p. 866).

(a) Show that one matrix square root of $$\Omega$$ is $$A = R A^{1/2}$$ where $$A^{1/2}$$ denotes another diagonal matrix whose diagonal elements are the square roots of the corresponding elements of A.

(b) Construct a second matrix square root with the additional property that it is symmetric.