CS223 · Annotated SlidesChapter 5 — Eigenvectors and Eigenvalues
Chapter 5 · Annotated slides

Eigenvectors and Eigenvalues

Eigenvectors Ax = λx, checking eigenvalues by solving (A − λI)x = 0, eigenspaces, eigenvalues of triangular matrices, and independence of eigenvectors for distinct eigenvalues.

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Slide 1: CS 223: Computational Linear Algebra
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    1. x ≠ 0 is an eigenvector of A with eigenvalue λ if Ax = λx. λ is an eigenvalue ⇔ (A − λI)x = 0 has a nontrivial solution.

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    The zero vector can be an eigenvector.

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        1. Eigenspace of λ = Nul(A − λI); its dimension = number of free variables.
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          Slide 7: Eigenvalues of triangular matrix
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          1. Eigenvalues of a triangular matrix are its diagonal entries. 0 is an eigenvalue ⇔ A is not invertible. Eigenvectors for distinct eigenvalues are linearly independent.
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          The eigenvalues of [[3, 6, −8], [0, 0, 6], [0, 0, 2]] are:

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