SINGULA

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DECOMPOSIT

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T I M L

I N

1

SINGULAR VALUE DECOMPOSITION

• Example 91. (page 128)

• Let A = . Then A’A = ,

• AA’ =

2

[2 1 11 2 11 1 2]

SINGULAR VALUE DECOMPOSITION

• The eigenvalues of A’A are 0, 1, 1, and 4; and the eigenvalues of AA’ are 1, 1, and 4.

• For l = 4, the eigenvectors of AA’ satisfy

• -2x + y + z = 0

• x – 2y + z = 0

• x + y -2z = 0

• A normalized solution (column eigenvector) is x = y = z =

• For l = 1, the eigenvectors of AA’ satisfy x + y + z = 0.

• There are two orthogonal normalized solutions (why two?) which are also orthogonal to the eigenvector of l = 4.

• x = , y = - , z = 0; and x = , y = , and z = .

3

SINGULAR VALUE DECOMPOSITION

• We have S’ = . Note the three columns of S’ were computed earlier as eigenvectors.

• From proof of theorem 9.2 (on page 126), U = A’S’L-1/2 = .=

4

SINGULAR VALUE DECOMPOSITION

• A SVD of A = .

•

5