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

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**preview**shows page 1. to view the full**5 pages of the document.**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 A∈IRm×n

r. Every A∈IRm×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 A∈IRm×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, A−L= [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 A∈IRm×n

r. Then

A+= lim

δ→0(ATA+δ2I)−1AT(1)

= lim

δ→0AT(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+=(α−1if α6= 0

0 if α= 0

Example 4: For any vector v∈IRn,

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 A∈IRm×nand suppose U∈IRm×m,V∈IRn×nare orthogonal (Mis

orthogonal if MT=M−1). Then

(UAV )+=VTA+UT.

Proof: Simply verify that the expression above does indeed satisfy each of the four Penrose

conditions.

Theorem: Let S∈IRn×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 A∈IRm×n,

1. A+= (ATA)+AT=AT(AAT)+

2. (AT)+= (A+)T

2

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