[
406
]
A GENERALIZED INVERSE FOR MATRICES
BY
R. PENROSE
Communicated by
J.
A.
TODD
Received
26 July 1954
This paper describes
a
generalization of the inverse of
a
nonsingular matrix, as the
unique solution of
a
certain set of equations. This generalized inverse exists for any
(possibly rectangular) matrix whatsoever with complex elements J. It is used here for
solving linear matrix equations, and among other applications for finding an expression
for the principal idempotent elements of a matrix. Also a new type of spectral decom
position is given.
In another paper
its
application
to
substitutional equations
and the
value
of
hermitian idempotents will be discussed.
Notation.
Capital letters always denote matrices (not necessarily square) with com
plex elements. The conjugate transpose of the matrix
A
is written
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*,
A
A*
=
0
implies
^4
= 0.
The last of these follows from the fact that the trace of
A
A*
is the sum of the squares
of the moduli of the elements of
A.
Erom the last two we obtain the rule
BAA*
=
CAA* implies BA
=
GA,
(1)
since
(BAA*CAA*)(BC)*
=
(BACA)(BAGA)*.
Similarly
BA*A
=
CA*A implies BA*
=
GA*.
(2)
THEOREM
\.
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.
I
first show that equations
(4) and
(5)
are
equivalent
to the
single equation
XX*A*
=
X.
(7)
Equation
(7)
follows from
(4) and
(5), since
it is
merely
(5)
substituted
in
(4).
Con
versely,
(7)
implies
AXX*A*
=
AX,
the
lefthand side
of
which
is
hermitian. Thus
(5) follows, and substituting (5)
in
(7) we get (4). Similarly, (3)
and
(6) can be replaced
by
the
equation
XAA*
= A*.
(8)
Thus
it
is sufficient
to
find
an
X
satisfying (7)
and
(8). Such
an
X
will exist
if a
B
can
be found satisfying
M
, ,
3
B
BA*AA*
= A*.
For then
X
=
BA*
satisfies
(8).
Also,
we
have seen that
(8)
implies
A*X*A*
=
A*
and therefore
BA*X*A*
=
BA*.
Thus
X
also satisfies
(7).
Now
the
expressions
A*A,
(A*A)
2
, (A*A)
3
,...
cannot
all be
linearly independent,
i.e. there exists
a
relation
A
1
A*A
+
\
2
(A*A)*+...+A
k
(A*A)
k
=
0
>
(9)
where A,,..., A
ft
are not all
zero.
Let
A,
be
the
first nonzero
A
and put
B
=

VH
Thus
(9)
gives
B(A*A)
r+1
=
(A*A)
T
,
and
applying
(1) and (2)
repeatedly
we
obtain
BA*AA*
=
A*,
as
required.
To show that
X
is unique, we suppose that
X
satisfies (7)
and
(8) and that
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 solution
of
(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 be
a
square matrix
and
may even
be
zero.)
I
shall also use
the
notation
A*
for
scalars, where
A
+
means A"
1
if
A
+
OandOif
A
= 0.
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 Spring '11
 Penrose
 Linear Algebra, Econometrics, Matrices, kx, A*, principal idempotent elements

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