Chapter 7, Problem 10SE

If A is m × 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 9 and 10.)

10. Show that if an n × n matrix G is positive semidefinite and has rank r, then G is the Gram matrix of some r × n matrix A. This is called a rank-revealing factorization of G. [Hint: Consider the spectral decomposition of G, and first write G as BB^T for an n × r matrix B.]