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A generalized inverse for matrices
R. Penrose
Mathematical Proceedings of the Cambridge Philosophical Society / Volume 51 / Issue 03 / July 1955, pp 406- 413DOI: 10.1017/S0305004100030401, Published online: 24 October 2008
Link to this article: http://journals.cambridge.org/abstract_S0305004100030401
How to cite this article:R. Penrose (1955). A generalized inverse for matrices. Mathematical Proceedings of theCambridge Philosophical Society, 51, pp 406-413 doi:10.1017/S0305004100030401
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[ 4 6
A GENERALIZED INVERSE FOR MATRICES
B Y R. PENROSE
Communicated by
J .
A. T O D D
Received26 July 1954
This paper describes
a
generalization of the inverse of
a
non-singular m atrix, as the
unique solution ofa certain set of equations. This generalized inverse exists for any
(possibly rectangu lar) m atr ix w hatsoever with complex elements J. I t is used here for
solving linear m atr ix eq ua tion s, an d among other applications for finding an expression
for the p rincipal idem po ten t elem ents of a m atrix. Also a new typ e of spectral decom-
position is given.
In another paper itsapplicat ion to substitutional equations and thevalueof
hermitian idempotents will be discussed.
Notation. Ca pital letter s alw ays deno te matrices (not necessarily square) with com-
plex elements. The conjugate transpose of the matrixAis writte n A*. Small letters
are used for column vectors (with an asterisk for row vectors) and small Greek letters
for complex numbers.
The following properties of the conjugate transpose will be used:
A** =A,
(A+B)* = A*+B*,
(BA)* = A*B*,
AA* =0implies
4
= 0.
Th e last of these follows from th e fact th a t the trace ofAA* is th e sum of the squ ares
of the moduli of the elements of
A.
Erom the last two we obta in the rule
BAA* = CAA* implies BA =G A, (1)
since (BAA*-CAA*)(B-C)* =(BA-CA)(BA-GA)*.
Similarly BA*A = CA*A implies BA* =GA*. (2)
T H E O R E M \.The four equations AX A A C\\
XAX =X, (4)
{AX)*
=AX, (5)
{XA)*
=XA, (6)
have a unique solution for any
A.
%
Matrices over more general rings will be considered in
a
later paper.
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A generalized inverse
for
matrices
407
Proof. Ifirst show that equations(4) and(5)areequivalentto thesingle equation
XX*A* =
X.
(7)
Equation (7)follows from (4) and(5), sinceit ismerely (5)substituted in(4). Con-
versely, (7)implies
AXX*A*
=
AX,
th eleft-hand sideofwhichishermitian. Thus
(5) follows, and substituting (5)
in
(7) we get (4). Similarly, (3)
and
(6) can be replaced
bytheequation XAA*
= A*.
(8)
Thusitis sufficient tofindan
X
satisfying (7)and(8). Suchan
X
will existif a
B
can
be found satisfying
M
, ,
3 B
BA*AA* = A*.
For then
X
=
BA*
satisfies (8).Also,wehave seen that (8)implies
A*X*A*
=
A*
and therefore
BA*X*A*
=
BA*.
Thus
X
also satisfies(7).
Nowtheexpressions
A*A,
(A*A)
2
, (A*A)
3
,... cannotall be linearly independent,
i.e. there existsarelation
A
1
A*A +\
2
(A*A)*+...+A
k
(A*A)
k
=0
>
(9)
where A,, ..., A
ft
are not allzero. Let
A,
bethefirst non-zero
A
and pu t
=
V
Thus
(9)
gives
B(A*A)
r+1
=
(A*A)
T
,
and
applying
(1) and (2)
repeatedly
we
obtain
BA*AA*
=
A*,
asrequired.
To show th at
X
is unique, we suppose that
X
satisfies (7)and(8) and tha t
Y
satisfies
Y
=
A*Y*Y
and
A* =A*AY.
These last relations are obtained by respectively
substituting (6)in(4)and(5)in
(3).
(They are (7)and(8) with
Y
in place of X and the
reverse order
of
multiplication and must,
by
symmetry, also
be
equivalent
to
(3),
(4),
(5)and(6).)Now
X = XX*A*
=
XX*A*AY = XAY
=
XAA*Y*Y = A*Y*Y
=
Y.
The unique solutionof
(3),
(4), (5)and(6) will be called the generalized inverse of
A
(abbreviated g.i.) and written
X =A* .
(Note that
A
need not beasquare matrixand
may evenbezero.) Ishall also usethenotation
A*
forscalars, where
A
+
means A
1
if
A
+OandOif
A
= 0.
In the calculation of
A*
it
is
only necessary
to
solve the
two
unilateral linear equations
XAA* =
A*
and
A*A Y = A*.
Byputting
A*
=
XA Y
andusingth efact that
XA
and
A Y
arehermitian andsatisfy
AX A = A
=
A YA
we observe that the four
relations
AA^A
=
A,
A'
t
AA
i
=
A*,
(AA
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4 0 8 R . P EN R O S E
Note that the properties
(A+B)* = A* +B*
and
(AA)*
=
A A*
were not used in
the proof of Theorem 1.
LEMMA.
1-1
A
n
= A;
1-2
A** = A**;
1-3 if A is
non-singular
A
f
= A-
1
;
1-4 (/U)
+
1-6
(A*Ay
=
1-6 ifUandV
are
un itary (UA V)
f
=
V*A
t
V*;
1-7 if A =~LA
it
where
A
i
A*
= 0 and ^ 4 * ^ = 0,
whenever
i+j then
A*
=-LAX;
1-8 i/
A is
normal
A
f
A
=
AA''
and
(A
n
)
f
= (
+
);
1-9 A, A*A,
A*
and A
i
A
all have rank equal to
traceA*A.
Proof. Since the g.i. of a matrix is always unique, relations 1-1,...,1-7 can be
verified by substituting the right-hand side of each in the denning relations for the
required g.i. in each case. For 1-5 we require (10) and for 1-7 we require the fact that
AiAlj = 0andA\A
}
=0ifi#j This follows fromA]=A?A}*A)andA\ = A\A\*A\.
The first p ar t of 1-8, of which the second pa rt is a consequence, follows from 1-5 and
(10), since
A*
A = (A*A)
f
A*A and AA
1
= (AA*)
f
AA*. (The condition for A to be
normal is
A*A =
AA*.)%To prove 1-9 we observe th at each of
A, A*A, A*
and
A
f
A
can be obtained from the others by pre- and post-multiplication by suitable matrices.
Thus their ranks must all be equal and
A''A
being idempotent has rank equal to its
trace.
Relation 1-5 provides a method of calculating the g.i. of a matrix A from the g.i.
of
A* A,
since
A*
=
(A*
Ay
A*.
Now
A* A,
being hermitian, canbereducedtodiagonal
form by a unitary transformation. HenceA*A = UDU*, where U is unitary and
D
= diag(a
1;
...,a
n
). Thus Z>
+
= diag(aj, . . . , < ) and (4*.4)
+
= UD*U*by 1-6.
An alterna tive proof of the existence of the g.i. of a square matrix can be obtained
from Autonne's
(1)
result that any square matrix
A
can be written in the form
VBW,
where Vand Ware unitary andBis diagonal . Now
B*
exists since it can be denned
similarly to D
f
above. It is thus sufficient to verify that A
f
=W*B*V* as in 1-6.
A rectangular m atrix can be trea ted by bordering it with zeros to make it square.
Notice thatA
f
is a continuous function ofAif the rank ofA is kept fixed , since in
th e singular case the polynomial in (9) could be taken to be the characteristic function
ofA*A. The value of
r,
being the nullity ofA *A(asA *Ais hermitian), is thus deter-
mined by the rank ofA (by 1-9). AlsoA
1
varies discontinuously when the rank ofA
changes since rankA = traceA*Aby 1-9.
J This result, like several of the earlier ones, is a simple consequence of the construction of
A* given in Theorem 1. It seems desirable, however, to show that it is also a direct consequence
of (3), (4), (5) and (6).
A simple proof of thi s resu lt is as follows: sinceAA* an d A*A are both hermitian and have
the same eigenvalues there exists a unitary matrix T such that
TAA*T*
= A*A. Hence TA is
nor ma l and therefore diagonable by uni tary transformation.
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A generalized inverse for matrices
409
THEOREM 2.
A necessary and sufficient condition for the equation AXB = C to have
a solution is AA
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410 R.
P E N R O S E
Proof.
The proofs of
2-1
an d 2-2 are eviden t. To prove
2-3
suppose
K
2
=K,
then, by
l-l,K = {(K*K) (KK')y.ThusE =KK'and F = Z'Zwilldo. Conversely,ifK = {FEf
t hen
K
is of the form
EFPEF
(sinceQ*
=
Q*(Q
f
*Q
f
Q
f
*)Q* by (10)). HenceK
=
EKF,
so tha t
K
2
= E(FEy FE{FEy
F
= E(FE)
f
F
=K.
Principal idempotent elementsofama trix.
For any square matrix
A
there exists
a unique set
of
m atricesK
x
denn ed for each complex num ber
A
such that
K
x
K^8
Xll
K
x
, (11)
ZK
X
=I,
(12)
AK
X
=K
X
A, (13)
(A-
A/)
K
x
is nilpotent, (14)
the non-zero
K
x
's
being the
principal idempotent
elements
of
A
(see Wedderburn(7))4
Unless Ais
an
eigenvalue ofA, K
x
is zero so that the sum
in
(12)isfinite, tho ug h
su m m ed over all complex A. Th e following theorem gives an explicit' formula forK
x
in terms ofg.i.'s.
T H B O B E M
3. / /
E
x
= I-{(A-U)
n
y(A-XI)
n
and F
x
= / -(A-U )
n
{ ( A - A/)
n
}
+
,
wheren issufficiently large (e.g. theorder of A), thenthe principal idempotent elements
ofA.are, given by K
x
= (F
x
E
x
y,andncanbetakenasunityifandonlyifAisdiagonable.
Proof.First, supposeA
is
diagonable and put
E
x
= I-(A-Aiy(A-M)
and F
x
=
I-(A-M)(A-Aiy.
ThenE
x
and F
x
are h .i. b y 2-1 and are zero unlessA is an eigenvalue ofA
by
1*3.
N o w
(AjuI)E =0 and F
x
{A-\I) =0, (15)
so tha t
[iF^p=F
x
AE
/t
=
A - F ^ . T hus
2 ^
= 0 if A*/*. (16)
Put t i ng
K
x
= (F
x
E
x
)
f
we have,by 2-3,
K
x
= E
x
(F
x
E
x
yF
x
. (17)
HenceK =S
X/l
K
x
b y (16). Also (17) gives
F^E, =
S^S^F.E^
(18)
Any eigenvectorz
a
oiAcorresponding to the eigenvalueasatisfies E
a
z
a
=z
a
. Since
A isdiagonable, an y colum n vecto rx conformable withA
is
expressible asasum of
eigenv ectors; i .e.itis expressible in the form
(a
finite sum over all com plex A)
x = ZE
x
x
x
.
| The existence of suchK
x
s can be established as follows:
let
4>(E)= (A^
1
.. .
( |
A^ '
be,
say,
the
minim um function
ofA
where
the
factors (
A
(
)
nl
are
mu tually coprime. Then
if
0()=(E,- A,)
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A generalized inverse
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411
Similarly,
if
y*
is
conformable with
A,i t
is expressible
as
Now 2/*(ig
x=
(Xy*
x
F
x
)
(S j
by (18) and (16). Hence
Eif^ = / .
Also, from (15) and (17) we obtain
AK
X
= XK
X
=
K
X
A. (19)
Thus conditions (13) and (14) are satisfied. Also
A
=
2AZ
A
.
(20)
Conversely,
if it is no t
assumed that
A is
diagonable but that
Y,K
X
= / ,K
x
being
defined as above (i.e. with
n= 1),
then
x=
HK
x
x gives
a;
as
a
sum of eigenvectors of
A,
since (19) was deduced without assuming the diagonability
ofA.
If
A
is not diagonable
i t
seems more convenient simply to prove tha t
for
any set of
K
x
s
satisfying (11), (12), (13) and (14) each
K
x
=
(F
x
E
x
y, whereF
x
and
E
x
are defined
as
in
the theorem.
Now,
if
the iC
A
's satisfy (11), (12), (13)
a nd
(14) they must certainly satisfy
and (A
- XI)
n
K
x
=
0 =
K
X
(A
-
A/) ,
where
n
is sufficiently large (ifnis the order
ofA,
this will do).
From this and the definitions of
E
x
andF
x
given in the theorem we obtain
E
X
K
X
= K
X
=K
X
F
X
. (21)
Using Euclid's algorithm we can find
P
and Q(polynomials
inA)
such that
I = (A
- M)
n
P
+Q(A - pi)*,
provided that A#/t. Now F
x
(A-XI)
n
= 0 =
(A-piy-E^. Hence F^E^
= 0 if
A=t=/t,
which with (21) givesF^K^
= 0 =
K^E whenever A#/t. But SZ
A
= / .
Thus
and K
X
E
X
= E
K
.\
( 2 2 )
Using (21)
and
(22)
i t is a
simple matter
to
verify that (-F
A
^
A
)
f
=
K
K
. (Note that
iT
A
=
/ - { ( / - #
A
) ( / - ^
A
) } t also.)
Incidentally, this provides
a
proof
of
the uniqueness
ofK
x
.%
COROLLARY.
/ /
A
is
normal,
it
isdiagonableand its principal idempotentelements
are
hermitian.
Proof.Since
A
XI
is
normal we
can
apply
1-8 to the
definitions
ofE
x
and
F
x
in
Theorem 3. This gives
E
X
= I-(A-XIY(A-XI) and F
x
=I -
(A
-
A/) (A
- XI)\
w h en ceA
is
diagonable. AlsoE
x
=
F
x
,so
t h a t K
x
=E
x
is
h e rmi t i an .
%
In
fact, S.K
A
=
I
and
(AXI)
n
K
x
= 0 are
sufficient
for
uniqueness.
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412 R. PBNROSE
Tor this result see, for example, Drazin(4).
If A is normal (20) givesA =AJ?
A
, and thus using 1-7, 1-4 and 2-2 we obtain
A new type of
spectral
decomposition.UnlessA is normal, there does not appear to be
any simple expression forA*in terms of its principal idempotent elements. However,
this difficulty is overcome with the following decomposition, which applies equally to
rectangular matrices.
T H E O R E M 4.Any matrix A is uniquely expressible in the form
A = 2 aU
a
,
this being a finite sum over real values of cc,where U\ = U*, U*U = 0and U
a
U^ =0
Proof.
The condition on the
U's
can be written comprehensively as
u
a
up
Y
= s
afi
s
fir
u
a
.
For
V
a
U*U
a
= U
a
implies TJ*JJ
a
U*
= 17*. whence by uniqueness,
TJ\
=
V*
(as
U
a
U*
and U U
a
are both hermitian). AlsoU
a
U Up=0 and U
a
U^ U =0 respectively imply
U* U
p
= 0 and U
a
U
fi
= 0 by (2). Define
E
x
= I - (A*A - Aiy (A*A - ?J).
The matrixA*A is normal, being hermitian, and is non-negative definite. Hence the
non-zero
JE^'s
are its principal idempotent elements and
JE7
A
= 0 unless A^ 0.
T h u s
A*A =2A^
A
and
(A*
Ay =2A^
A
.
Hence A*A = [A*Ay A*A =SA
+
A^
A
= S E
x
.
A
Put U
a
= otr^AE ifa>0 and U
a
=
0
otherwise. Then
a
A>0
Also
if a, y > 0 U
a
U*
fi
U
y
= u-
1
fi-
1
y-
1
AE
at
E
pt
A*AE
yt
To prove the uniqueness suppose
A = 2
8
fi7
V
a
.
a>0
Then it is easily verified that the non-zero expressions V V
a
, fora >0, and / 2 l
7
?^
are the principal idempotent elements of A*A corresponding respectively to the
eigenvalues a
2
and 0. Hence
V*V
a
= E
a
2, where oc> 0, giving
tf. = *-H 2 / ^ ) F*F
a
=V
a
.
Polar representation of a matrix.Itis a well-known result (closelyrelatedto Autonne's)
that any square matrix is the product of a hermitian with a unitary matrix (see, for
example, Halmos 5)).
| ThusA* =
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A generalized inverse for matrices 413
P u t
H =
2af/
a
U .
Then
H
is non-negative definite hermitian, since
U
a
=
0 unless
a>0, and H
2
=
AA*. (Such an H must be unique.) We have WH = HW = ^44
+
.
Now AA
i
an d ^4
+
JL are equivalent und er a un itary transforma tion, both being
herm itian and h aving the same eigenvalues; i .e. there is a unitary m atrix
W
satisfying
WA*A = AA'W. Put t ing V= H*A + W- WA*Aw e see th at VV* = I andA = HV.
This is the
polar
representationof
A.
The polar representation is unique if A is non-singular but not if A is singular.
How ever, if we require
A = HU,
where i? is(asbefore) non-negative definite herm itian,
U
f
= U* and UU* = WH, the represen tation is always un iqu ej and also exists for
recta ng ular ma trices. The uniqueness of If follows from
AA*
=
HUU*H
=
H(Hm)H = H
2
,
and that of
U
from
WA
=
WHU
=
UU*U
=
U.
The existence of
H
and J7 is established by
H
= Sa?7
a
Z7*an d J7 = SC/
a
.
If we put G =SaC/
a
*U
a
we obtain the alternative representation A = [/(?.
I wish to th an k D r M. P . Drazin for useful suggestions and for advice in the prepa ra-
tion of this paper.
REFERENCES
(1) AUTONNB, L. Sur les matrices hypohermitiennes et sur les matrices unitaires. Ann. Univ.
Lyon,
(2), 38 (1915), 1-77.
(2)
BJEBHAMMAB,
A. Rec tangu lar reciprocal matr ices, with special reference to geodetic calcula-
tions.
Bull.
gdod.
int. (1951), pp. 188-220.
(3)
CECIONI,
F . Sopra operazioni algebriche. Ann. Scu. norm. sup. Pisa, 11 (1910), 17-20.
(4) DBAZTN,M. P . On diagonable and norm al matrices. Quart. J. Math.(2), 2 (1951), 189-98.
(5) HAIMOS, P. R. Finite dimensionalvector spaces (Princeton, 1942).
(6) MURRAY, F. J. and VON NEUMANN,J. On rings of operators. Ann. Math., Princeton,(2),
37 (1936),
141-3.
(7) WEDDERBTJBN,J. H. M. Lectures onmatrices (Colloq. Publ. Amer. m ath .
Soc.
no. 17, 1934).
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