Polynomials, Invariant Subspaces, and Eigenvalues

Sets up the algebraic tools eigenvalue theory depends on — zeros of a polynomial, the division algorithm, and factorization over $\mathbf{C}$ (every polynomial splits) versus over $\mathbf{R}$ (irreducible quadratic factors survive) — then defines eigenvalues via invariant subspaces, applies polynomials to operators, and proves existence of an eigenvalue for every operator on a complex vector space via the minimal polynomial, which also gives the sharper existence result for odd-dimensional real vector spaces.

Outline (TODO — flesh out each)

• Zeros of polynomials; division algorithm
• Factorization of polynomials over $\mathbf{C}$
• Factorization of polynomials over $\mathbf{R}$
• Invariant subspaces and eigenvalues
• Polynomials applied to operators
• Existence of eigenvalues on complex vector spaces
• The minimal polynomial; eigenvalues and the minimal polynomial
• Eigenvalues on odd-dimensional real vector spaces