Maths 3.5Lesson outlineLesson 5 of 7 in Module 3
Eigenvalues and Eigenvectors
Find the directions a matrix only stretches, and read the eigenvalues as how much.
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In this lesson11 parts
- 01The directions a matrix only stretches
- 02Arrows that turn and arrows that do not
- 03Reading the eigenvalues
- 04Not every matrix has them
- 05The eigendecomposition
- 06Eigenvectors as the axes of contours
- 07Checkpoint: eigenvectors so far
- 08The used car lot's loss as a quadratic
- 09Why gradient descent zigzags
- 10Eigenvectors in NumPy
- 11Recap, challenge, and next: the singular value decomposition
Key terms
The words this lesson introduces, each in one line. The module’s glossary collects them all.
- eigenvector
- a nonzero vector the matrix only stretches,
Av = λv|(1, 1) - eigenvalue
λ, the factor it is stretched by |3- symmetric matrix
- equal to its own transpose; real eigenvalues, perpendicular eigenvectors |
XᵀX - eigendecomposition
A = QΛQᵀ: rotate, stretch along the axes, rotate back |Λ = diag(1, 3)- quadratic form
xᵀAx, a quadratic built from a symmetric matrix |2x² + 2xy + 2y²
Quiz
Your first pick on each question is the one that counts, and a right one earns a coin. Getting one wrong here is how the lesson sticks.