Singular Values

Here's the picture of for another matrix .

The plot on the left shows a grid spanned by an orthonormal basis .

The matrix sends this grid to an orthogonal grid on the right spanned by .

The grid on the left has squares, but the grid on the right has rectangles.

To make the grid on the right have the same scale as the one on the left add unit vectors in the direction of .

Here's the same picture of with unit vectors and in the directions of and .

The matrix sends

to the direction of and

to the direction of .

Since you see that stretches vectors in the direction of by a factor of .

Since you see that stretches vectors in the direction of by a factor of .

The numbers and are called the singular values of the matrix .

Here's the picture of for another matrix .

The graphic shows that sends and to the directions of and .

Move to to compute .

Move to to compute .

(Q11) The singular values of are __ and __.

You can use the singular values of to compute .

will stretch by 3 and send it to .

will stretch by and send it to .

So .

(Q12) Move to to verify .

Here's the picture of for another matrix .

The matrix sends and to the directions of and .

This time the vector is stuck at .

(Q13) The graphic shows

.

How much did stretch ? How much did stretch ?

(Q14) The singular values of are __ and __.

(Q15) Use the singular values of to compute .

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)