15. Singular Values
σ1 =
p
λ1 =
√
12 (7)
σ2 =
p
λ2 =
√
10 (8)
Singular Values Decompostion of A
Σ =
"√
12 0 0
0
√
10 0
#
5/18
16. Constructing Matrix U
A = UΣV T
(9)
AAT
= UΣV T
(UΣV T
)T
(10)
= UΣT
V T
V ΣUT
(11)
= UΣ2
UT
(12)
= UDUT
(13)
AAT
=
"
11 1
1 11
#
(14)
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17. Constructing Matrix U Cont’d
When λ = 12
"
11 − λ 1
1 11 − λ
#
=
"
−1 1
1 −1
#
(15)
By row reduction form, we have:
"
−1 1
1 −1
#
=>
"
−1 1
0 0
#
(16)
Forming equations with some variables:
−x + y = 0 (17)
Our eigenvector becomes: (1, 1)t
7/18
18. Constructing Matrix U Cont’d
When λ = 10
"
11 − λ 1
1 11 − λ
#
=
"
1 1
1 1
#
(18)
By row reduction form, we have:
"
1 1
1 1
#
=>
"
1 1
0 0
#
(19)
Forming equations with some variables:
x + y = 0 (20)
Our eigenvector becomes: (1, −1)t
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19. Constructing Matrix U Cont’d
Let Z be the egigenvectors of the various eigenvalues.
Z =
"
1 1
1 −1
#
We normalized each column of Z to get U
U =
"√
2
2
√
2
2
√
2
2 −
√
2
2
#
9/18