Singular Value Decomposition
If a matrix stretches and sends an orthonormal basis to an orthonormal basis then you can compute as a three step process.
The singular value decomposition of writes as a product of three matrices where matrix does step .
Three steps to compute .
Step 1: Find the coordinates for .
Since is an orthonormal basis the -coords for are .
(1) See "orthonormal coordinates" in Getting Started.
Choose to be the matrix with rows .
(2) See "row way" in Getting Started.
Step 2: Multiply the coordinates by the singular values and .
Choose the matrix .
Step 3: Use as the coordinates of .
Choose to be the matrix with columns .
(3) See "column way" in Getting Started.
Combining all three steps you get
The singular value decomposition for is
(Q19) In terms of the vectors and , the singular value decomposition for is .
Copyright © 2007 Todd Will
Created by Mathematica (April 15, 2007)