University Linear Algebra Eigenvalues and Diagonalization — Unit Notes
Every definition and theorem this unit expects you to know, stated once each and written to be revised from: 4 definitions and 10 theorems, 7 worked examples, 1 diagram, in 13 printable pages. This page lists what the notes cover. The notes themselves come with the University Linear Algebra bundle.
Unit notes — in the bundle
Not sold separately. The practice worksheet for this unit is free, with no email and no account.
Eigenvalues and Diagonalization — what the notes cover
8 sections, in the order the unit is taught. Each entry below is the title of a definition or a theorem the notes state in full.
What you need
Also in this section: 1 note on method and common errors.
Eigenvalues and eigenvectors
- Definition — Eigenvector and eigenvalue
- Theorem — Eigenvalues and the null space
Also in this section: 1 worked example, 1 note on method and common errors, 1 diagram.
The characteristic polynomial
- Definition — Characteristic polynomial and algebraic multiplicity
- Theorem — Eigenvalues are the roots
- Theorem — Eigenvalues of a triangular matrix
- Theorem — Trace and determinant from the eigenvalues
Also in this section: 1 worked example, 1 note on method and common errors.
Eigenspaces and their dimensions
- Definition — Eigenspace
- Theorem — Dimension of an eigenspace
Also in this section: 1 worked example, 1 note on method and common errors.
Diagonalizing a matrix
- Definition — Diagonalizable matrix and operator
- Theorem — The diagonalization theorem
Also in this section: 1 worked example, 1 note on method and common errors.
Deciding whether a matrix is diagonalizable
- Theorem — Distinct eigenvalues give independent eigenvectors
- Theorem — The test for diagonalizability
Also in this section: 1 worked example, 1 note on method and common errors.
Powers of a diagonalizable matrix
- Theorem — Powers through a diagonalization
Also in this section: 1 worked example, 1 note on method and common errors.
Complex eigenvalues of a real matrix
- Theorem — Eigenvalues of a real matrix
Also in this section: 1 worked example, 1 note on method and common errors.
Where the notes are
The notes are one of the documents in the University Linear Algebra Solutions Bundle, beside the unit test, the challenge problems and every worked answer key. The Eigenvalues and Diagonalization practice worksheet is free and stays free.
Every University Linear Algebra topic — the complete Solutions Bundle
One download, one payment, the whole program. For all 9 University Linear Algebra units: the worksheet, the reference notes, the challenge set, the unit test and every answer key — these notes included.
- Worked solutions, not answer lists — every step written out
- Covers the whole year's program at this level
- Less than the price of one hour of tutoring — for the entire year's solutions
Practise it, then test yourself
Notes on their own do not make anything stick. This unit has a free, printable practice worksheet, and an end-of-unit test that comes with the bundle.
Eigenvalues and Diagonalization practice worksheet About the Eigenvalues and Diagonalization unit test