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

COURSE DETAILS

Level:

Type:

Undergraduate

(A-)


Instructors: Nikolaos Marmaridis, Ioannis Beligiannis
Department: Department of Mathematics
Institution: University of Ioannina
Subject: Mathematics
Rights: CC-BY-SA

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