Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem VIII.1 (Rank-one modifications) Let X = I – axx^*, where x is a column vector, α is a scalar, and I is the identity matrix.

1. (a) Show that XX^* = I – βxx^*, for some real number β.
2. (b) Consider a matrix Y of the form Y = I – βxx^*, for some real scalar β. For what condition on β is y positive semi-definite? When this is the case, show that Y admits a Hermitian square-root factor of the form Y^1/2 = I – axx^* for some real scalar α.

Problem VIII.2 (Basis rotation) In this problem we provide another proof for Lemma 33.1 by resorting to the QR decomposition of a matrix rather than its singular value decomposition. The SVD proof given in the text is more general since the argument in this problem assumes that the matrices A and B have full rank. Introduce the QR decompositions

where Q_A and Q_B are M × M unitary matrices, and R_A and R_B are n × n upper triangular matrices with positive diagonal entries (due to the full rank assumption on A and B).

1. (a) Show that
2. (b) Conclude, by uniqueness of the Cholesky factorization (cf. Sec. B.3), that R_A = R_B. Com plete the argument to show that the matrix unitary and maps B to A

Problem VIII.3 (Sample ...