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.

Get the book

The Pseudoinverse: the Shortest Solution

Chapter 8 · Mathematics for AI and Machine Learning · Advanced · 1:46

An interactive live film: the solutions of two equations in three unknowns form a line parallel to the null space; the pseudoinverse picks the one in the row space, where the smallest sphere touches the line, and ridge solutions slide to it. Find the shortest solution.

Narration

Two equations, three unknowns: x one plus x two equals two, and x two plus x three equals one. Each equation is a plane, and the two planes meet in a line. Every point on the green line solves both.

The line runs parallel to the null space of A, the dashed line through the origin. Adding any null vector to a solution gives another solution.

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

More from Chapter 8

Interactive1:49SVD: Rotate, Stretch, RotateChapter 8Preview1:31Energy from Singular Values: Rebuild a Picture Layer by LayerChapter 8Preview1:33Double Descent at the Interpolation ThresholdChapter 8Preview1:32The Marchenko-Pastur Law and the Escaping EigenvalueChapter 8Preview1:43Eckart-Young: the Best Low-Rank ApproximationChapter 8Preview1:31Polar Decomposition: Stretch, then RotateChapter 8Preview1:47Randomized SVD: Catching the Top DirectionsChapter 8