BIZNIQUE / LINEAR ALGEBRA / 18
Interpret an eigenvector geometrically
Interpret an eigenvector geometrically
Look for a direction that a matrix stretches without turning away from its span.
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THE IDEA
Read the example.
Read the example.
An eigenvector remains on the same line after transformation; its eigenvalue gives the signed scale factor. For the symmetric example A = [[3, 1], [1, 3]], the diagonal directions have eigenvalues 4 and 2.
Av = λv, for v ≠ 0
Use it well
Use it well
Eigenvectors can reverse direction when the eigenvalue is negative. General real matrices need not have a complete real eigenbasis.
TRY IT YOURSELF
A quick check.
A quick check.
What happens to v = (1, 1) under this example matrix?
Check your reasoning
Check your reasoning
Av = (4, 4) = 4v, so v is an eigenvector with eigenvalue 4.
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