By Ali H. Sayed

ISBN-10: 0470253886

ISBN-13: 9780470253885

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We then define its inertia as the triplet In{M} = {I+, I - , l o } , where I+ ( M ) = the number of positive eigenvalues of M I- ( M ) = the number of negative eigenvalues of M I o ( M ) = the number of zero eigenvalues of M Congruence. An important result regarding congruent matrices is the following, which states that congruence preserves inertia. , for any Hermitian M and invertible Q , it holds that I n { M } = In{QMQ*}. Thus, assume that M has the block Hermitian form A B* D ] with and consider the corresponding block factorizations A=A*, D = D* 17 where now A D = A - BD-'B' AA = D - B*A-'B, SECTION B 4 CHOLESKY FACTORIZATION The above factorizations have the form of congruence relations so that we must have In(M) = In ([ A 0 AA I) and In(M) = I n ([ I) Positive-definite matrices.

N - l}, of dimensions N x 1 each with N 2 n. These vectors can be converted into an equivalent set of orthonormal vectors { q L . i = 0,1.. . , n - l}. which span the same linear subspace as the {hi}, by appealing to the classical Gram-Schmidt procedure. This is an iterative procedure that operates as follows. It starts with 40 = ho//(hoIl,where ilholl denotes the Euclidean norm of ho, and then repeats for i > 0: i-1 Thus, observe, for example, that T I = hl - (qGhl)qo, which is simply the residual vector that results from projecting hl onto qo.

Assume not. Then there exists a nonzero vector p such that Ap = 0. It follows that A*Ap = 0, which contradicts the invertibility of A* A. This is because A*Ap = 0 implies that p is an eigenvector of A * A corresponding to the zero eigenvalue. Hence, the determinant of A* A is necessarily zero. Finally, let us show that A'A is positive-definite. For this purpose, take any nonzero vector I and consider ~ . the product I'A'Az is necessarily positive; it cannot the product x*A*Ax,which evaluates to l l A ~ l /Then, be zero since the nullspace of A , in view of A being full rank, contains only the zero vector.

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Adaptive filters MNw by Ali H. Sayed

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