Inverse of A

Suppose where and are orthonormal bases for .

To solve is a three step process.

If is invertible you can write where does all three steps at once.

You can also write as a product where matrix does step .

Step 1: Find coords of . Since is an orthonormal basis the coords are .

Choose . Then .

Step 2: Divide the numbers by the singular values.

Choose .

Then .

Step 3: Use the entries of as the coords of .

Choose .

Then .

Combining all three steps, the solution to is

.

If is invertible, then the solution to is just . So it's a safe bet that the product is really just .

(Q43) Suppose .

Express as a product of three matrices.

Of course since step 2 divides by the singular values, this procedure only works if the none of the singular values is zero. What happens if ???

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Copyright © 2007 Todd Will

Created by
Mathematica (April 15, 2007)