eigenvalues and eigenvectors
(2.1 hours to learn)
Summary
If A is a square matrix, the eigenvalues are the scalar values u satisfying Ax = ux, and the eigenvectors are the values of x. Eigenvectors and eigenvalues give a convenient representation of matrices for computing powers of matrices and for solving differential equations. An important special case is the spectral decomposition of symmetric matrices.
Context
This concept has the prerequisites:
- matrix multiplication (Eigenvectors and eigenvalues can be used to analyze repeated matrix multiplications.)
- roots of polynomials (The eigenvalues can be computed as roots of a polynomial.)
- linear systems as matrices (Eigenvalues are defined as solutions to a system of linear equations.)
- complex numbers (The eigenvalues and eigenvectors may be complex valued.)
- determinant (The characteristic polynomial, used for computing eigenvalues analytically, is given in terms of a determinant.)
- column space and nullspace (Eigenspaces can be viewed as nullspaces of a particular matrix.)
Goals
- Know the definitions of eigenvalues and eigenvectors
- Understand why the eigenvectors for a given eigenvalue form a subspace
- Be able to calculate eigenvalues and eigenvectors in terms of the roots of the characteristic polynomial
- Show that the sum of the eigenvalues equals the trace
- Show that the product of the eigenvalues equals the determinant
- Know why eigenvalues and eigenvectors can sometimes be complex valued
- Show that the complex eigenvalues of a real matrix come in conjugate pairs
- Show that eigenvectors corresponding to distinct eigenvalues are linearly independent
Core resources (read/watch one of the following)
-Free-
→ A First Course in Linear Algebra (2012)
A linear algebra textbook with proofs.
Additional dependencies:
- complex vectors and matrices
→ MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Location:
Lecture "Eigenvalues and eigenvectors"
-Paid-
→ Introduction to Linear Algebra
An introductory linear algebra textbook with an emphasis on numerical methods.
Location:
Section 6.1, "Introduction to eigenvalues," pages 283-291
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location:
Section 9.2, "Eigenvalues, eigenvectors, and applications," up to "Diagonalizability," pages 422-429
Additional dependencies:
- change of basis
Supplemental resources (the following are optional, but you may find them useful)
-Free-
→ Khan Academy: Linear Algebra
See also
- Eigenvalues are closely related to the characteristic polynomial of a matrix.
- The determinant of a matrix is the product of the eigenvalues .
- The Spectral Theorem states that symmetric matrices have a full set of eigenvalues with orthogonal eigenvectors.
- Eigenvalues are used in solving linear ordinary differential equations .