(Singular Value Decomposition) Let A be a real symmetric matrix. Let B be a full-column rank matrix such that Col(B) = Col(A) so that the columns of B are a basis for the columns of A.

(a) Given A, how could you find such a B?

(b) Show that $$A = BC$$ where C has the same dimensions as B.

(c) Show that there is a nonsingular matrix D so that $$C = BD$$. [HINT: Show that Col(B) = Col(C).]

(d) Hence, show that A can always be decomposed into $$A = BHB'$$ where H is nonsingular and symmetric and B is full-column rank. Such decompositions are called singular value decompositions.

(e) Use this result to propose a matrix square root for A if it is also positive semidefinite.