eigenvalues and eigenvectors
(2.1 hours to learn)
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.
This concept has the prerequisites:
- 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)
→ A First Course in Linear Algebra (2012)
→ MIT Open Courseware: Linear Algebra (2011)
Videos for an introductory linear algebra course focusing on numerical methods.
Location: Lecture "Eigenvalues and eigenvectors"
→ 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
- change of basis
Supplemental resources (the following are optional, but you may find them useful)
→ Khan Academy: Linear Algebra
Location: Lecture sequence "Eigen-everything"
- 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 .
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