If A is m × n, then the matrix G = ATA 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 BBT for an n × r matrix B.]
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