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Unformatted text preview: ECEN 601: Assignment 8 Problems: 1. Let V be a finite-dimensional inner product space and T a linear operator on V . Show that the range of T * is the orthogonal complement of the null space of T . 2. Let V be the space of n × n matrices over the complex numbers, with the inner product ( A | B ) = tr( AB * ). Let P be a fixed invertible matrix in V , and let T P be the linear operator on V defined by T P ( A ) = P- 1 AP . Find the adjoint of T P . 3. Let V be a finite-dimensional inner product space, and let E be an idempotent linear operator on V , i.e., E 2 = E . Prove that E is self-adjoint if and only if EE * = E * E . 4. If P is a projection matrix, show that I − P is a projection matrix. Determine the range and nullspace of I − P . 5. Show that B = I − A ( A H A )- 1 A H is positive semidefinite, and hence that the minimum error e min = x − P A x = B x has an equal or smaller norm than the original vector x . Hint: Consider bardbl e min bardbl 2 ≥ 0....
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This note was uploaded on 09/26/2011 for the course ECEN 601 taught by Professor Staff during the Fall '08 term at Texas A&M.
- Fall '08