Chapter 6 - Matrices

Abstract

After a short introduction on the definition and elementary properties of matrices and determinants, we introduce the partitioning of matrices, the different special matrices (null, diagonal, scalar, identity, symmetric, complex conjugate, real, transpose, inverse, adjoint, Hermitian, orthogonal and unitary matrices) and the matrix eigenvalue problem, as well as more advanced matrix techniques. Among these we introduce the functions of Hermitian matrices, the analytic functions, the construction of matrix projectors and the canonical form, the matrix pseudoeigenvalue problem, the Lagrange interpolation formula and the Cayley–Hamilton theorem. The applications include the diagonalization of the Hückel matrix for the π electron system of the benzene molecule, and many worked-out examples, including the calculation of the inverse, square root and exponential of Hermitian matrices, the square root of a non-symmetric matrix having degenerate eigenvalues, and the complete solution of the eigenvalue and pseudoeigenvalue problems for a (2×2) Hermitian matrix.