Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem II.1 (Rank-one modification of the identity matrix) Consider any matrix of the form I + xy^T, where x and y are column vectors. Use the matrix inversion formula (5.4) to show that its inverse is also a rank-one modification of the identity. More specifically, show that

Probiem II.2 (Determinant Of a matrix) Consider a square matrix A. A fundamental result in matrix theory is that every such matrix admits a so-called canonical Jordan decomposition, which is of the form A = UJU⁻¹, where J = diag{J₁, …, J_r} is a block diagonal matrix, say with r blocks. Each J_i is bi-diagonal with identical diagonal entries λ_i and with unit entries along the lower diagonal, namely,

The size of each J_i say n_i × n_i, indicates the multiplicity of the eigenvalue λ_i. Show that det .

Remark. When A is Hermitian, it can be shown that J is necessarily diagonal (rather than block diagonal with bi-diagonal blocks).

Problem 11.3 (Trace of a matrix) Use the canonical Jordan factorization of Prob. II.2, and the fact that Tr(XY) = Tr(YX) for any matrices {X, Y} of compatible ...