Properties Moore–Penrose inverse

1 properties

1.1 existence , uniqueness
1.2 basic properties

1.2.1 identities


1.3 reduction hermitian case
1.4 products
1.5 projectors
1.6 geometric construction
1.7 subspaces
1.8 limit relations
1.9 continuity
1.10 derivative

properties

proofs of these facts may found on separate page proofs involving moore–penrose inverse.

existence , uniqueness

the pseudoinverse exists , unique: matrix



a




{\displaystyle a\,\!}

, there precisely 1 matrix




a

+






{\displaystyle a^{+}\,\!}

, satisfies 4 properties of definition.

a matrix satisfying first condition of definition known generalized inverse. if matrix satisfies second definition, called generalized reflexive inverse. generalized inverses exist not in general unique. uniqueness consequence of last 2 conditions.

basic properties

if



a




{\displaystyle a\,\!}

has real entries,




a

+






{\displaystyle a^{+}\,\!}

.
if



a




{\displaystyle a\,\!}

invertible, pseudoinverse inverse. is:




a

+


=

a

−
1






{\displaystyle a^{+}=a^{-1}\,\!}

.
the pseudoinverse of 0 matrix transpose.
the pseudoinverse of pseudoinverse original matrix:



(

a

+



)

+


=
a




{\displaystyle (a^{+})^{+}=a\,\!}

.
pseudoinversion commutes transposition, conjugation, , taking conjugate transpose:








(

a


t




)

+


=
(

a

+



)


t



,
 
 
(



a
¯




)

+


=



a

+


¯


,
 
 
(

a

∗



)

+


=
(

a

+



)

∗


.




{\displaystyle (a^{\mathrm {t} })^{+}=(a^{+})^{\mathrm {t} },~~(\,{\overline {a}}\,)^{+}={\overline {a^{+}}},~~(a^{*})^{+}=(a^{+})^{*}.\,\!}






the pseudoinverse of scalar multiple of reciprocal multiple of a:








(
α
a

)

+


=

α

−
1



a

+






{\displaystyle (\alpha a)^{+}=\alpha ^{-1}a^{+}\,\!}





α
≠
0.


{\displaystyle \alpha \neq 0.}





identities

the following identities can used cancel subexpressions or expand expressions involving pseudoinverses. proofs these properties can found in proofs subpage.

a

+




=



a

+





a

+
∗





a

∗







a

+




=



a

∗





a

+
∗





a

+






a


=



a

+
∗





a

∗




a




a


=


a



a

∗





a

+
∗







a

∗




=



a

∗




a



a

+







a

∗




=



a

+




a



a

∗








{\displaystyle {\begin{array}{lclll}a^{+}&=&a^{+}&a^{+*}&a^{*}\\a^{+}&=&a^{*}&a^{+*}&a^{+}\\a&=&a^{+*}&a^{*}&a\\a&=&a&a^{*}&a^{+*}\\a^{*}&=&a^{*}&a&a^{+}\\a^{*}&=&a^{+}&a&a^{*}\\\end{array}}}





reduction hermitian case

the computation of pseudoinverse reducible construction in hermitian case. possible through equivalences:

a

+


=
(

a

∗


a

)

+



a

∗






{\displaystyle a^{+}=(a^{*}a)^{+}a^{*}\,\!}







a

+


=

a

∗


(
a

a

∗



)

+






{\displaystyle a^{+}=a^{*}(aa^{*})^{+}\,\!}

as




a

∗


a


{\displaystyle a^{*}a}

,



a

a

∗




{\displaystyle aa^{*}}

hermitian.

products

if



a
∈

m

(
m
,
n
;
k
)
,
 
b
∈

m

(
n
,
p
;
k
)



{\displaystyle a\in \mathrm {m} (m,n;k),~b\in \mathrm {m} (n,p;k)\,}

, , if

a




{\displaystyle a\,\!}

has orthonormal columns (i.e.,




a

∗


a
=

i

n





{\displaystyle a^{*}a=i_{n}\,}

), or




b




{\displaystyle b\,\!}

has orthonormal rows (i.e.,



b

b

∗


=

i

n





{\displaystyle bb^{*}=i_{n}\,}

), or




a




{\displaystyle a\,\!}

has columns linearly independent (full column rank) ,



b




{\displaystyle b\,\!}

has rows linearly independent (full row rank), or




b
=

a

∗






{\displaystyle b=a^{*}\,\!}

(i.e.,



b


{\displaystyle b}

conjugate transpose of



a


{\displaystyle a}

),

then

(
a
b

)

+


≡

b

+



a

+






{\displaystyle (ab)^{+}\equiv b^{+}a^{+}\,\!}

.

the last property yields equivalences:

(
a

a

∗



)

+





≡

a

+
∗



a

+






(

a

∗


a

)

+





≡

a

+



a

+
∗








{\displaystyle {\begin{aligned}(aa^{*})^{+}&\equiv a^{+*}a^{+}\\(a^{*}a)^{+}&\equiv a^{+}a^{+*}\end{aligned}}}



projectors

p
=
a

a

+






{\displaystyle p=aa^{+}\,\!}

,



q
=

a

+


a




{\displaystyle q=a^{+}a\,\!}

orthogonal projection operators – is, hermitian (



p
=

p

∗






{\displaystyle p=p^{*}\,\!}

,



q
=

q

∗






{\displaystyle q=q^{*}\,\!}

) , idempotent (




p

2


=
p




{\displaystyle p^{2}=p\,\!}

,




q

2


=
q




{\displaystyle q^{2}=q\,\!}

). following hold:

p
a
=
a
q
=
a




{\displaystyle pa=aq=a\,\!}

,




a

+


p
=
q

a

+


=

a

+






{\displaystyle a^{+}p=qa^{+}=a^{+}\,\!}






p




{\displaystyle p\,\!}

orthogonal projector onto range of



a




{\displaystyle a\,\!}

(which equals orthogonal complement of kernel of




a

∗






{\displaystyle a^{*}\,\!}

).




q




{\displaystyle q\,\!}

orthogonal projector onto range of




a

∗






{\displaystyle a^{*}\,\!}

(which equals orthogonal complement of kernel of



a




{\displaystyle a\,\!}

).




(
i
−
q
)
=
(
i
−

a

+


a
)




{\displaystyle (i-q)=(i-a^{+}a)\,\!}

orthogonal projector onto kernel of



a




{\displaystyle a\,\!}

.




(
i
−
p
)
=
(
i
−
a

a

+


)




{\displaystyle (i-p)=(i-aa^{+})\,\!}

orthogonal projector onto kernel of




a

∗






{\displaystyle a^{*}\,\!}

.

the last 2 properties imply following identities:

a

 
(
i
−

a

+


a
)
=
(
i
−
a

a

+


)
a
 
 
=
0


{\displaystyle a\,\ (i-a^{+}a)=(i-aa^{+})a\ \ =0}







a

∗


(
i
−
a

a

+


)
=
(
i
−

a

+


a
)

a

∗


=
0


{\displaystyle a^{*}(i-aa^{+})=(i-a^{+}a)a^{*}=0}

another property following: if



a
∈

r

n
×
n




{\displaystyle a\in r^{n\times n}}

hermitian , idempotent (true if , if represents orthogonal projection), then, matrix



b
∈

r

m
×
n




{\displaystyle b\in r^{m\times n}}

following equation holds:

a
(
b
a

)

+


=
(
b
a

)

+




{\displaystyle a(ba)^{+}=(ba)^{+}}

this can proven defining matrices



c
=
b
a


{\displaystyle c=ba}

,



d
=
a
(
b
a

)

+




{\displaystyle d=a(ba)^{+}}

, , checking



d


{\displaystyle d}

indeed pseudoinverse



c


{\displaystyle c}

verifying defining properties of pseudoinverse hold, when



a


{\displaystyle a}

hermitian , idempotent.

from last property follows that, if



a
∈

r

n
×
n




{\displaystyle a\in r^{n\times n}}

hermitian , idempotent, matrix



b
∈

r

n
×
m




{\displaystyle b\in r^{n\times m}}

(
a
b

)

+


a
=
(
a
b

)

+




{\displaystyle (ab)^{+}a=(ab)^{+}}

finally, should noted if



a


{\displaystyle a}

orthogonal projection matrix, pseudoinverse trivially coincides matrix itself, i.e.




a

+


=
a




{\displaystyle a^{+}=a\,\!}

.

geometric construction

if view matrix linear map



a
:

k

n


→

k

m




{\displaystyle a:k^{n}\to k^{m}}

on field



k


{\displaystyle k}






a

+


:

k

m


→

k

n




{\displaystyle a^{+}:k^{m}\to k^{n}}

can decomposed follows. write



⊕


{\displaystyle \oplus }

direct sum,



⊥


{\displaystyle \perp }

orthogonal complement,



ker


{\displaystyle \operatorname {ker} }

kernel of map, ,



ran


{\displaystyle \operatorname {ran} }

image of map. notice




k

n


=
(
ker
⁡
a

)

⊥


⊕
ker
⁡
a


{\displaystyle k^{n}=(\operatorname {ker} a)^{\perp }\oplus \operatorname {ker} a}

,




k

m


=
ran
⁡
a
⊕
(
ran
⁡
a

)

⊥




{\displaystyle k^{m}=\operatorname {ran} a\oplus (\operatorname {ran} a)^{\perp }}

. restriction



a
:
(
ker
⁡
a

)

⊥


→
ran
⁡
a


{\displaystyle a:(\operatorname {ker} a)^{\perp }\to \operatorname {ran} a}

isomorphism. these imply




a

+




{\displaystyle a^{+}}

defined on



ran
⁡
a


{\displaystyle \operatorname {ran} a}

inverse of isomorphism, , on



(
ran
⁡
a

)

⊥




{\displaystyle (\operatorname {ran} a)^{\perp }}

zero.

in other words: find




a

+


b


{\displaystyle a^{+}b}

given b in k, first project b orthogonally onto range of a, finding point p(b) in range. form a({p(b)}), i.e. find vectors in k sends p(b). affine subspace of k parallel kernel of a. element of subspace has smallest length (i.e. closest origin) answer




a

+


b


{\displaystyle a^{+}b}

looking for. can found taking arbitrary member of a({p(b)}) , projecting orthogonally onto orthogonal complement of kernel of a.

this description closely related minimum norm solution linear system.

subspaces









ker
⁡
(

a

+


)



=
ker
⁡
(

a

∗


)




ran
⁡
(

a

+


)



=
ran
⁡
(

a

∗


)






{\displaystyle {\begin{aligned}\operatorname {ker} (a^{+})&=\operatorname {ker} (a^{*})\\\operatorname {ran} (a^{+})&=\operatorname {ran} (a^{*})\end{aligned}}}



limit relations

the pseudoinverse limits:







a

+


=

lim

δ
↘
0


(

a

∗


a
+
δ
i

)

−
1



a

∗


=

lim

δ
↘
0



a

∗


(
a

a

∗


+
δ
i

)

−
1




{\displaystyle a^{+}=\lim _{\delta \searrow 0}(a^{*}a+\delta i)^{-1}a^{*}=\lim _{\delta \searrow 0}a^{*}(aa^{*}+\delta i)^{-1}}


(see tikhonov regularization). these limits exist if



(
a

a

∗



)

−
1






{\displaystyle (aa^{*})^{-1}\,\!}

or



(

a

∗


a

)

−
1






{\displaystyle (a^{*}a)^{-1}\,\!}

not exist.

continuity

in contrast ordinary matrix inversion, process of taking pseudoinverses not continuous: if sequence



(

a

n


)


{\displaystyle (a_{n})}

converges matrix (in maximum norm or frobenius norm, say), (an) need not converge a. however, if matrices have same rank, (an) converge a.

derivative

the derivative of real valued pseudoinverse matrix has constant rank @ point



x


{\displaystyle x}

may calculated in terms of derivative of original matrix:

d



d

x




a

+


(
x
)
=
−

a

+



(



d



d

x



a
)


a

+


 
+
 

a

+



a

+

t




(



d



d

x




a

t


)


(
i
−
a

a

+


)

 
+
 

(
i
−

a

+


a
)


(


d


d

x




a

t


)


a

+

t




a

+




{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}a^{+}(x)=-a^{+}\left({\frac {\mathrm {d} }{\mathrm {d} x}}a\right)a^{+}~+~a^{+}a^{+{\text{t}}}\left({\frac {\mathrm {d} }{\mathrm {d} x}}a^{\text{t}}\right)\left(i-aa^{+}\right)~+~\left(i-a^{+}a\right)\left({\frac {\text{d}}{{\text{d}}x}}a^{\text{t}}\right)a^{+{\text{t}}}a^{+}}






^ cite error: named reference gvl1996 invoked never defined (see page).
^ stoer, josef; bulirsch, roland (2002). introduction numerical analysis (3rd ed.). berlin, new york: springer-verlag. isbn 978-0-387-95452-3. .
^ anthony a. maciejewski , charles a. klein, obstacle avoidance kinematically redundant manipulators in dynamically varying environments . international journal of robotics research, 1985.
^ rakočević, vladimir (1997). on continuity of moore–penrose , drazin inverses (pdf). matematički vesnik. 49: 163–172. 
^ http://www.jstor.org/stable/2156365