Linear Algebra II

Sum and direct sum of subspaces. The polynomial rings R[t] and C[t]. Eigenvalues, eigenvectors, and eigenspaces. Diagonalization and triangulation of endomorphisms and square matrices. Minimal polynomial and the Cayley-Hamilton Theorem. Applications to linear recurrent sequences, computation of powers and inverse of a matrix. Symmetric bilinear forms and inner products. Euclidean spaces. Orthonormal bases and the Gram-Schmidt orthogonalization process. Orthogonal subspaces and orthogonal complements. Orthogonal matrices and isometries. Adjoint of an endomorphism and of a square matrix. Self-adjoint endomorphisms and symmetric matrices. The spectral theorem for self-adjoint endomorphisms (Euclidean case) and symmetric matrices. Geometric interpretation of isometries. Positive and non-negative endomorphisms. Norm of a matrix. Quadratic forms and principal axes. Classification of quadratic surfaces. Hermitian spaces. Hermitian and unitary matrices. The spectral theorem for self-adjoint endomorphisms (Hermitian case) and Hermitian matrices.

Objectives

Prerequisites

Linear Algebra I

Syllabus