The defining equation
A nonzero v is an eigenvector of A when Av=λv. The reflection about sends (x,y) to (−y,−x).
Algebra / English
Eigenvalues and Eigenvectors Intuitive
chrisimm1 · YouTube · 6:25
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Read the eigenvectors and eigenvalues of a reflection from its fixed and reversed directions.
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1:05The defining equation3:44The fixed direction5:20The reversed directionLearning script
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A nonzero v is an eigenvector of A when Av=λv. The reflection about sends (x,y) to (−y,−x).
The vector (1,−1) lies along the reflection line and is unchanged. Its eigenvalue is 1.
The vector (1,1) maps to (−1,−1), so its eigenvalue is −1. These two independent directions form an eigenbasis for this example; a general n-by-n matrix need not have n distinct real eigenvalues.
Knowledge cards
The fixed direction
The vector (1,−1) lies along the reflection line and is unchanged. Its eigenvalue is 1.
The reversed direction
The vector (1,1) maps to (−1,−1), so its eigenvalue is −1. These two independent directions form an eigenbasis for this example; a general n-by-n matrix need not have n distinct real eigenvalues.
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- Eigenvalues and eigenvectors ExplanationAt 1:05
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A nonzero v is an eigenvector of A when Av=λv. The reflection about sends (x,y) to (−y,−x). The vector (1,−1) lies along the reflection line and is unchanged. Its eigenvalue is 1. The vector (1,1) maps to (−1,−1), so its eigenvalue is −1. These two independent directions form an eigenbasis for this example; a general n-by-n matrix need not have n distinct real eigenvalues.
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