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Unformatted text preview: s the fact that for a unitary matrix U , U � U = U U � = I , we have AA + �
Σ−1 0
=U
VV
U�
0
0
�
�
� � −1
Σ0
Σ
0
=U
I
U�
00
0
0
�
�
Ir×r 0
=U
U �,
0
0
� Σ0
00 � 4 � � which is symmetric. Similarly,
� Σ−1 0
0
0 � Σ−1
0 � + Ir×r
0 AA = V
=V
=V �
Σ0
V�
UU
00
�
��
0
Σ0
V�
I
0
00
�
0
V�
0
� � � which is again symmetric. The facts derived above can be used to show the other two. + AA A =
=
=
= � Ir×r
(AA )A = U
0
�
�
�
Ir×r 0
�
UU
U
0
0
�
�
Σ0
U
V�
00
A.
+ 0
0 � U �A
�
Σ0
V�
00 Also, + + A AA �
Ir×r 0
= (A A)A = V
V � A+
0
0
�
�
�
� −1
Ir×r 0
Σ
0
�
=V
U�
VV
0
0
0
0
� −1
�
Σ
0
=V
U�
0
0
+ � + = A+ .
b) We have to show that when A has full column rank then A+ = (A� A)−1 A� , and that when A
has full row rank then A+ = A� (AA� )−1 . If A has full column rank, then we know that m ≥ n,
rank (A) = n, and
�
�
Σn×n
A=U
V �.
0
Also, as shown in chapter 2, when A has full column rank, (A� A)−1 exists. Hence
�
� �−1
�
�
�
��
��
Σ
�
−1 �
(A A) A =
V Σ 0 UU
V�
V Σ� 0 U �
0
�
�
��
�
�
� −1
V Σ 0 U�
= V Σ ΣV
�
�
= V (Σ� Σ)−1 V � V Σ� 0 U �
�
�
= V (Σ� Σ)−1 Σ� 0 U �
= V ( Σ−1 0 )U �
= A+ .
5 Similarly, if A has full row rank, then n ≥ m, rank (A) = m, and
�
�
A = U Σm×m 0 V � .
It can be proved that when A has full row rank, (A� A)−1 exists. Hence,
� � −1 A (AA ) �
� � � �−1
�
��
Σ
=V
U U Σ 0 VV
U�
0
� ��
�
�− 1
Σ
=V
U � U ΣΣ� U �
0
� ��
Σ
=V
U � U (ΣΣ−1 )U �
0
� −1 �
Σ
=V
U�
0
� Σ�
0 � � = A+ .
c) Show that, of all x that minimize �y − Ax�2 , the one with t...
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This document was uploaded on 03/19/2014 for the course EECS 6.241J at MIT.
 Spring '11
 EmilioFrazzoli
 Computer Science, Electrical Engineering

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