Eigenvalues and Eigenvectors

Eigenvalues, eigenvectors, and the characteristic polynomial are defined, first in the context of real matrices. Later, the complex numbers are introduced in order to prove the Spectral Theorem for real, symmetric matrices and Singular Value Decomposition (SVD). Students have the opportunity to calculate eigenvalues and eigenvectors of 3x3 matrices in the associated software. The concepts of algebraic and geometric multiplicity of eigenvalues of real matrices are treated, as is the topic of diagonalization, including a proof that a real matrix is diagonalizable if and only if there exists a basis of eigenvectors. The text concludes with an introduction to complex numbers and the Fundamental Theorem of Algebra, which are used to fully prove the Spectral Theorem. SVD is proved as an application of the Spectral Theorem and ties together nearly all of the topics of the text.