Unit 5 of 10

Spectral Theory

  • 5 lessons
  • Videos 22 to 26

This unit asks which operators can be simplified by an orthonormal change of basis, and how far that simplification can go.

Invariant subspaces give block triangular matrices, and Schur's theorem shows that every complex square matrix is upper triangular in some orthonormal basis, with a real version that uses diagonal blocks of size one and two. Over the complex numbers, normal operators are exactly the ones with an orthonormal eigenbasis. This gives the spectral theorem for Hermitian and unitary operators, and a finite family of commuting normal operators shares one orthonormal eigenbasis.

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.

Invariant Blocks and Schur Form

  1. Invariant Subspaces and Block Matrices

    What makes a subspace invariant under an operator, how an adapted basis turns invariance into a block upper triangular matrix, when the matrix becomes block diagonal, and what a reducing subspace adds.

    Video 22

  2. Schur Decomposition

    Why every complex square matrix is upper triangular in some orthonormal basis, built from a unit eigenvector, a unitary matrix, a compression and induction on dimension, and how its diagonal gives the trace and determinant.

    Video 23

  3. Real Schur Form

    How a conjugate eigenpair of a real matrix gives a real invariant plane, why real Schur form uses diagonal blocks of size one and two, and when a two-by-two block is a rotation-scaling block.

    Video 24

Normal Operators and Simultaneous Diagonalization

  1. The Spectral Theorem for Normal Operators

    Why a complex operator has an orthonormal eigenbasis exactly when it is normal, how unitary diagonalization works, and why Hermitian eigenvalues are real and unitary eigenvalues have modulus one.

    Video 25

  2. Simultaneous Diagonalization

    When several operators share one orthonormal eigenbasis, why commuting operators preserve each other's eigenspaces, how normal restrictions to reducing subspaces finish the proof, and why commuting is not enough.

    Video 26