Praise for Fundamentals of WiMAX

168 Chapter 5 • Multiple-Antenna TechniquesSidebar 5.1 A Brief Primer on Matrix TheoryAs this chapter indicates, linear algebra and matrix analysis are an inseparable part ofMIMO theory. Matrix theory is also useful in understanding OFDM. In this book, wehave tried to keep all the matrix notation as standard as possible, so that any appropriatereference will be capable of clarifying any of the presented equations.In this sidebar, we simply define some of the more important notation for clarity.First, in this chapter, two types of transpose operations are used. The first is the conventionaltranspose A T , which is defined asTA ij=A jithat is, only the rows and columns are reversed. The second type of transpose is the conjugatetranspose, which is defined as*A ij=( ) * .A jiThat is, in addition to exchanging rows with columns, each term in the matrix isreplaced with its complex conjugate. If all the terms in A are real, A T = A * . Sometimes,the conjugate transpose is called the Hermitian transpose and denoted as A H . They areequivalent.Another recurring theme is matrix decomposition—specifically, the eigendecompositionand the singular-value decomposition, which are related to each other. If a matrix issquare and diagonalizable (M × M), it has the eigendecompositionA = TΛT –1 ,where T contains the (right) eigenvectors of A, and Λ = diag[λ 1 λ 2 ... λ M ] is a diagonalmatrix containing the eigenvalues of A. T is invertible as long as A is symmetric or hasfull rank (M nonzero eigenvalues).When the eigendecomposition does not exist, either because A is not square or forthe preceding reasons, a generalization of matrix diagonalization is the singular-valuedecomposition, which is defined asA = UΣV * ,where U is M × r, V is N × r, and Σ is r × r, and the rank of A—the number of nonzerosingular values—is r. Although U and V are no longer inverses of each other as in eigendecomposition,they are both unitary—U*U = V*V = UU* = VV* = I—which meansthat they have orthonormal columns and rows. The singular values of A can be related toeigenvalues of A*A byσ i ( A) = λ i ( A*A).Because T –1 is not unitary, it is not possible to find a more exact relation between the singularvalues and eigenvalues of a matrix, but these values generally are of the same order,since the eigenvalues of A*A are on the order of the square of those of A.