Unit 8 of 10

Canonical Forms

  • 8 lessons
  • Videos 38 to 45

This unit studies operators that cannot be diagonalized and builds the Jordan form, which describes a complex operator completely up to similarity.

Polynomials in an operator lead to the minimal polynomial, the Cayley-Hamilton theorem and an exact test for diagonalizability, and cyclic subspaces show how a single vector generates an invariant subspace. Polynomial projectors split the space into generalized eigenspaces, and chains of generalized eigenvectors inside each one give a Jordan basis. The unit ends by reading the block sizes from kernel dimensions and by showing why exact Jordan form is unreliable when the entries are only known approximately.

Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Polynomial Tools and Cyclic Structure

  1. Matrix Polynomials and the Minimal Polynomial

    How to evaluate a polynomial at an operator, why some nonzero polynomial annihilates it, how division shows that the minimal polynomial divides every annihilating polynomial, and why its roots in the field are the eigenvalues.

    Video 38

  2. The Cayley-Hamilton Theorem

    Why every square matrix satisfies its characteristic polynomial, proved from the adjugate identity and polynomials with matrix coefficients, how this reduces high powers, and how characteristic and minimal polynomials differ.

    Video 39

  3. Polynomial Projectors and Diagonalizability

    How the Euclidean algorithm and Bézout's identity turn coprime factors into polynomial projectors, why diagonalizability means a minimal polynomial with distinct linear factors, and when commuting operators share an eigenbasis.

    Video 40

  4. Cyclic Subspaces and Companion Matrices

    How the orbit of one vector spans a cyclic subspace, why its first dependence gives an orbit basis, a companion matrix and a linear recurrence, and when an operator has a cyclic vector.

    Video 41

Chains and Generalized Eigenspaces

  1. Generalized Eigenvectors

    Generalized eigenvectors and Jordan chains for operators with too few eigenvectors, why every chain is independent, why the nested kernels of powers stabilize, and the generalized eigenspace.

    Video 42

  2. Generalized Eigenspace Decomposition

    Why the space is the direct sum of the generalized eigenspaces of an operator whose minimal polynomial splits, proved with polynomial projectors, with nilpotent restrictions and dimensions equal to algebraic multiplicities.

    Video 43

Jordan Form Construction and Computation

  1. Constructing Jordan Form

    How to build a Jordan basis for a nilpotent operator from nested bases chosen top down and lifted to chains, why the chains are independent and span, and how Jordan blocks assemble into the Jordan form.

    Video 44

  2. Computing Jordan Form and Its Limitations

    How successive kernel dimensions count the Jordan blocks of each size, how to compute and check a Jordan basis, why the block sizes are unique, and why a tiny perturbation can change the Jordan form.

    Video 45