12.5. THE SINGULAR VALUE DECOMPOSITION∗
229
This proves the corollary. The singular value decomposition also has a very interesting connection to the problem of least squares solutions. Recall that it was desired to find x such that |Ax − y| is as small as possible. Lemma 12.3.2 shows that there is a solution to this problem which can be found by solving the system A∗ Ax = A∗ y. Each x which solves this system solves the minimization problem as was shown in the lemma just mentioned. Now consider this equation for the solutions of the minimization problem in terms of the singular value decomposition. A∗
A
A∗
z µ }| ¶ {z µ }| ¶ { z µ }| ¶ { σ 0 σ 0 σ 0 ∗ ∗ V U U V x=V U ∗ y. 0 0 0 0 0 0 Therefore, this yields the following upon using block multiplication and multiplying on the left by V ∗ . µ 2 ¶ µ ¶ σ 0 σ 0 V ∗x = U ∗ y. (12.13) 0 0 0 0 One solution to this equation which is very easy to spot is µ −1 ¶ σ 0 x=V U ∗ y. 0 0
(12.14)