Canonical Forms

What replaces diagonalization when an operator isn’t diagonalizable: every operator on a complex vector space decomposes the space into generalized eigenspaces (kernels of powers of $T - \lambda I$), on each of which $T$ is a scalar plus a nilpotent operator — this is the Jordan form. Covers nilpotent operators and generalized eigenvectors, the generalized eigenspace decomposition, block-diagonal matrices, the trace as an invariant connecting matrices and operators, and the rational canonical form as the field-independent analogue when eigenvalues don’t exist in the base field.

Outline (TODO — flesh out each)

• Null spaces of powers of an operator; generalized eigenvectors
• Nilpotent operators
• Generalized eigenspace decomposition; multiplicity of an eigenvalue
• Block diagonal matrices
• Square roots of operators
• The Jordan canonical form (I and II)
• The minimal polynomial, revisited via canonical forms
• Trace — a connection between matrices and operators
• The rational canonical form