The interactive film is in the book's Companion

Play it, pause it, turn it and drag it with Mathematics for AI and Machine Learning.

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Polar Decomposition: Stretch, then Rotate

Chapter 8 · Mathematics for AI and Machine Learning · Advanced · 1:31

An interactive live film: three layers show the unit circle, its stretch by S = V Sigma V^T, and the rigid turn by Q = U V^T that gives A. Drag A's columns until Q is the identity.

Narration

A complex number is a length times a turn, r times e to the i theta. Every square matrix has the same kind of factorization: A equals Q times S.

The bottom layer is the unit circle, with colored points on it. The top layer is what A does to them: an ellipse, with every point moved somewhere new.

The rest of the narration is in the book's Companion.

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