17. Let T and U be a self-adjoint linear operators on an n-dimensional inner product space V, and let A = [T]a, where B is an orthonormal basis for V. Prove the following results. (a) T is positive definite [semidefinite] if and only if all of its eigenval- ues are positive [uonnegative] (b) T is positive definite if and only if E Aijajai 0 for all nonzero n-tuples (aj . (2. ... . (In). i. j (c) T is positive semidefinite if and only if A = B*B for some square matrix B. (d) If T and U are positive semidefinite operators such that T2 = U2, then T = U. (e) If T and U are positive definite operators such that TU = UT. then TU is positive definite. (f) T is positive definite [semidefinite] if and only if A is positive def- inite [semidefinite]. Because of (f). results analogous to items (a) through (d) hold for ma- trices as well as operators. 378 Chap. 6 Inner Product Spaces 18. Let T: V - W be a linear transformation, where V and W are finite- dimensional inner product spaces. Prove the following results. (a) T*T and TT* are positive semidefinite. (See Exercise 15 of Sec- tion 6.3.) (b) rank(T*T) = rank(TT*) = rank(T)