If A is m x n, then the matrix G = A^TA is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 25 and 26.)

Show that if an n x n matrix G is positive semidefinite and has rank r, then G is the Gram matrix of some r x n matrix A. This is called a rank-revealing factorization of G.