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 .

Then .

(2) See "row way" in Getting Started.

Step 2: Multiply the coordinates by the singular values and .

Choose the matrix .

Then .

Step 3: Use as the coordinates of .

Choose to be the matrix with columns .

Then .

(3) See "column way" in Getting Started.

Combining all three steps you get

.

The singular value decomposition for is

.

Here's the picture of for another matrix .

(Q19) In terms of the vectors and , the singular value decomposition for is .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)