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MATH 33AH Lecture Notes - Lecture 18: A Aa, Diagonal Matrix, Unmanned Aerial VehicleExam


Department
Mathematics
Course Code
MATH 33AH
Professor
All
Study Guide
Final

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The Moore-Penrose Pseudoinverse (Math 33A: Laub)
In these notes we give a brief introduction to the Moore-Penrose pseudoinverse, a gen-
eralization of the inverse of a matrix. The Moore-Penrose pseudoinverse is defined for any
matrix and is unique. Moreover, as is shown in what follows, it brings great notational
and conceptual clarity to the study of solutions to arbitrary systems of linear equations and
linear least squares problems.
1 Definition and Characterizations
We consider the case of AIRm×n
r. Every AIRm×n
rhas a pseudoinverse and, moreover,
the pseudoinverse, denoted A+IRn×m
r, is unique. A purely algebraic characterization of
A+is given in the next theorem proved by Penrose in 1956.
Theorem: Let AIRm×n
r. Then G=A+if and only if
(P1) AGA =A
(P2) GAG =G
(P3) (AG)T=AG
(P4) (GA)T=GA
Furthermore, A+always exists and is unique.
Note that the above theorem is not constructive. But it does provide a checkable cri-
terion, i.e., given a matrix Gthat purports to be the pseudoinverse of A, one need simply
verify the four Penrose conditions (P1)–(P4) above. This verification is often relatively
straightforward.
Example: Consider A="1
2#. Verify directly that A+= [1
5,2
5]. Note that other left
inverses (for example, AL= [3 ,1]) satisfy properties (P1), (P2), and (P4) but not (P3).
Still another characterization of A+is given in the following theorem whose proof can
be found on p. 19 in Albert, A., Regression and the Moore-Penrose Pseudoinverse, Aca-
demic Press, New York, 1972. We refer to this as the “limit definition of the pseudoinverse.”
Theorem: Let AIRm×n
r. Then
A+= lim
δ0(ATA+δ2I)1AT(1)
= lim
δ0AT(AAT+δ2I)1(2)
1
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2 Examples
Each of the following can be derived or verified by using the above theorems or characteri-
zations.
Example 1: A+=AT(AAT)1if Ais onto, i.e., has linearly independent rows (Ais right
invertible)
Example 2: A+= (ATA)1ATif Ais 1-1, i.e., has linearly independent columns (Ais left
invertible)
Example 3: For any scalar α,
α+=(α1if α6= 0
0 if α= 0
Example 4: For any vector vIRn,
v+= (vTv)+vT=(vT
vTvif v6= 0
0 if v= 0
Example 5: "1 0
0 0 #+
="1 0
0 0 #
This example was computed via the limit definition of the pseudoinverse.
Example 6: "1 1
1 1 #+
="1
4
1
4
1
4
1
4#
This example was computed via the limit definition of the pseudoinverse.
3 Some Properties
Theorem: Let AIRm×nand suppose UIRm×m,VIRn×nare orthogonal (Mis
orthogonal if MT=M1). Then
(UAV )+=VTA+UT.
Proof: Simply verify that the expression above does indeed satisfy each of the four Penrose
conditions.
Theorem: Let SIRn×nbe symmetric with UTSU =D, where Uis orthogonal and D
is diagonal. Then S+=UD+UTwhere D+is again a diagonal matrix whose diagonal
elements are determined according to Example 3.
Theorem: For all AIRm×n,
1. A+= (ATA)+AT=AT(AAT)+
2. (AT)+= (A+)T
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