Equations for inversion of singular matrix - special form of generalized inverse of arbitrary complex matri

Decell, H. P., Jr.

English

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N A S A  TECHNICAL NOTE 
A SPECIAL FORM OF A 
GENERALIZED INVERSE OF A N  
ARBITRARY COMPLEX MATRIX 
by Henry Po Decell, Jr. 
Manned Spacecraft Center 
Houston, Texas 
N A T I O N A L  AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  WASHINGTON,  D. C. APRIL  1 9 6 5  
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I lll lll lll I Ill IllI llll 1111 
A SPECIAL FORM OF A GENERALIZED INVERSE 
O F  AN ARBITRARY COMPLEX MATRIX 
By Henry P. Decell, Jr. 
Manned Spacecraft Center 
Houston, Texas 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - Price $1.00 
A SPECIAL FORM OF A GENERALIZED INVERSE 
OF AN ARE3ITRARY COMPLEX MATRIX 
By Henry P. Decell, Jr. 
Manned Spacecraft Center 
SUMMARY 
T'he primary concern of this paper is to investigate the problem of inver- 
sion of singular or nonsquare matrices. 
for computing the generalized inverse of an arbitrary complex matrix is given. 
For a nonsingular matrix the algorithm gives the ordinary inverse of the matrix. 
In this connection, a new algorithm 
The paper is divided into several sections. The first two sections give 
a definition-theorem expose' of the known results in the literature. 
lowing sections give a new explicit form, together with an algorithm for com- 
puting the new explicit form. 
is given that can easily be realized in trajectory analysis problems. 
a computer program for computing the generalized inverse of a matrix is given 
utilizing the algorithm mentioned in the latter paragraph. 
The fol- 
An application to least squares approximation 
Finally, 
INTRODUCTION 
A. Bjerhammar (ref. l), E. H. Moore (ref. 2), and R. Penrose (ref. 3 )  
independently generalized the concept of matrix inversion to include arbitrary 
complex matrices. Their equivalent forms of the generalized inverse of a 
matrix have given rise to many applications of generalized inversion. 
text, the basic theory is utilized in giving a new explicit form of the gener- 
alized inverse of an arbitrary complex matrix. 
In the 
SYMBOLS 
Capital letters 
Lowercase letters 
A" 
A=& 
matrices unless otherwise stated 
column vectors unless otherwise stated or clear from 
context 
matrix conjugate transpose of A 
4 I 
I 
I 
matrix inverse for nonsingular A 
A+ 
H 
pR(A) 
( 
Em 
an> 
diag al,a2, . . ., 
generalized inverse of A 
a Hermitian idempotent matrix (h.i..); that is, a matrix 
such that IF = H and HK = H 
range space of A; that is, the collection of all images 
of column vectors under the transformation A 
orthogonal projection on the range of A; that is, a 
Hermitian idempotent leaving R(A) fixed 
m-dimensional euclidean space 
diagonal matrix 
DEFINITIONS AND EQUIVA"T FORMS 
A. Bjerhammar (ref. l), E. H. Moore (ref. 2), and R. Penrose (ref. 3 )  
independently generalized the concept of matrix inversion\to include arbitrary 
complex matrices, The generalized inverse of a singular, or nonsquare, matrix 
possesses properties which make it a central concept in matrix theory. 
In this, paper, a definition-theorem expose is presented, along with 
applicable references and special problems. 
due to Penrose (ref. 3) is stated without proof: 
The following fundamental theorem 
THEOREM I. The four equations 
(Ax)" = A x  (3) 
(XA)" =xA (4) 
have a unique solution X for each complex matrix A. 
Definition 1. The solution X in THEOREM 1 is denoted X = A+ and is 
The following theorem gives an equivalent form of A*. 
called the generalized inverse of A. 
THEOREM 11. For,any mxm matrix A over the complex field, X = A+ 
and XA = P 
R(X) is the unique solution to the equations Ax = P R(A) 
2 
where R(A) is the range space of A in E" and P is the orthogonal 
projection on R(A). 
R(A) 
Proof: THEOREM I implies that AX is a Hermitian idempotent (see Symbols) 
which leaves A fixed, that is, (AX)A = A. Hence, AX must be a projection. 
It may be concluded that XA is a lso  a projection. 
The properties of the generalized inverse and possible computing schemes 
we. given in the following theorem. 
THEOREM 111. Let A be an arbitrary complex matrix. Then, for scalar 
A # o and unitary u and V, 
(A+)+ = A ( 7 )  
(A*)+ = (A+)* (8) 
A+ = A=' for nonsirqgiiar A ( 9 )  
+ + *  (A*A)+ = A (A 
(12) 
-1 + 1 (UAV)' = V A U- 
J 
3 
* * 
If A is normal (i.e., A A = AA ) then 
A'A = AA+ and (An)+ = (A+)n (14) 
* +  + A, A A, A , and A A all have rank equal to 
trace (A+A) (15) 
* + *  A+ = (A A )  A 
+ Equation (16) reduces the problem of computing A 
the generalized inverse of a Hermitian matrix A A. Moreover, such a mtrix 
can always be diagonalized by a unitary transformation, that is, 
to that of computing 
9 
* 
D = U(A A ) V  = diag(a 1 , ..., an) 
Equations (10) and (12) imply that 
i t  
( A  A)' = VD'U = V diag(i1,.. .,+ n)U
It is tacitly assumed that if a = 0, the corresponding term in i 
diag(i1,.. .,$ ) is zero. It is not usually an easy task to determine the 
unitary transformations U and V. Methods for computing the generalized 
inverse have been given by various authors (refs. 1, 4, 5, 6, and 7). 
n 
The following is a theorem of major importanze characterizing all solL: 
tions of the matrix equations AXB = C which have some solution X. 
THEOREM IV. For the matrix equation AXB = C to have a solution, a 
necessary and sufficient condition is 
+ +  A A C B B = C  
irl which case, the general solution is 
+ +  x = A CB + Y - A+AYBB+ 
where Y is arbitrary to within the limits of being consistent with the di- 
mension in the indicated multiplications (ref. 3) . 
4 
Proof: If X satisfies AXB = C, 
+ +  c = AXB = .AA+AXBB+B = AA CB B 
+ +  Conversely, if c = AA CB B, A+CB+ is a particular solution. Clearly, for 
the general solution, AXB = 0 must be solved. Any expression of the form 
x = Y - A+AYBB+ 
The only property required of A+ and B+ in the theorem is AA'A = A and 
E B B = B .  + 
Corollary A. The general solution to the vector equation 
F!%=c 
is 
+ x = P c + (I - P+?)Y 
where y is arbitrary, provided a solution exists. 
Corollary B. A necessary and sufficient condition for the equations 
A x = c  
and 
X B = D  
to have a common solution is that each have a solution and AD = CB (ref. 8). 
Proof: If AX = C and XB = D have a common soluticn, then clearly each has 
a solution and 
A x B = c B  
A X B = A D  
so that 
CB ='AD 
In order to obtain the sufficiency of the condition, it is assumed that 
x = A+C + DB+ - A+ADB+ 
which is a solution if AD = CB, A'C = C, and DB' = D. 
5 
THEOREM V. 
+ A'A, A', I - A A, and I - A' are h.i. 
H is h.i. implies H+ = H 
(See Symbols. ) 
Proof: The proof requires a straightforward application of THEOREN I. 
In general, the reversal rule (i.e., (AB)' = B+A+ as in the case of' the 
standard inverse) does not hold. 
following result: 
R. Cline (ref. 9 )  recently obtained the 
THEOREM VI. Let A and B be matrices with the product AB defined. 
+ +  Then, 
(AB)' = BIAl 
where : 
AB = AIB1 
and 
THE EXPLICIT FORM 
Utilizing the properties of A+ in the preceding sections, an explicit 
form is developed which gives rise to an algorithm for computing the generalized 
inverse of an arbitrary complex matrix (ref. 5). 
THEOREM VII. For any matrix A, A+ = WAY where W and Y are any 
solutions of 
(19)  
* 
WAA* = A  
and 
3 * 
A A Y = A  
Proof: Equations (19) and (20) indeed have a solution W = Y = A+. Moreover, 
if W. and Y are any solutions, 
* * * 
AWAA" = AA and A AYA = A A 
so that 
AWA = A and AYA = A 
6 
* * 
Note: BAA = CAA implies BA = CA 
In addition, 
+ *  * *  + *  * *  
WAAW = A W  and Y A A Y = Y A  
* imply 
(WA)* = WA and (AY) = AY 
If it is assumed that X = WAY, X satisfies equations (1) to (4) of 
THEOREM I SO that A+ = X = WAY. 
* * 
Corollary C. For any matrix A, A+ = A S1AS2A where S1 and S2 are, 
respectively, any solutions of 
(AA*)S1(AA*) = (a*) 
and 
(A*A)s,(A*A) = (A + A) 
* * 
Proof: According to THEOREM 111, W = A S1 and Y = S2A are solutions of 
equations (19) and (20) of THEOREM V I 1  provided 
(AA*)S,o() = (AA*) 
and 
* * + 
(A A)S2(A A) = (A A) 
The corollary follows. 
THEOREM VIII. If B is a matrix and nonsingular matrices P and Q 
exist so that PBQ = E is an idempotent, then B = QEP is a solution of 
BXB = B. 
Proof: If P, Q, and E satisfy the hypothesis of the theorem, 
and 
Corollary C and THEOREM VI11 suggest an algorithm for computing the 
generalJzed inverse of a complex matrix F. Consider the equation 
F+ =I (F F) F + *  + from reference 10, which reduces the problem of finding F 
7 
* 
to that of finding the generalized inverse of the Hermitian matrix F F = C. 
2 Since (C2)* = C , nonsingular matrices P and Q exist (products of elemen 
tary matrices obtained by simple. elimination) so t k t  
is a rank r identity mtrix and the Z is a zero matrix. When Ir where 
C is set equal to A in Corollary C, 
According to THEOREM VIII, solutions S = S2 = QI P are chosen so that 
1 0 
2 c+ = (csl) c 
+ (F*F)+ = c 
and finally, 
Computing programs f r c  
+ c  
F + = C F  
1 culat ing Sl and S2 are now in existence 
(e.g., STORM, Statistically Oriented kbtrix Program, IBM) . 
programs only compute some solution of the equation 
+ from A . These results allow one to construct a solution to all four Penrose 
equations (eqs. (1) to (4) of THEOREM I), given only a solution of the first, 
namely, AXA = A. 
In general, these 
AXA = A, usually different 
APPLICATION TO LEAST SQUARES APPROXIMATION 
In this section, an application to the least squares approximation is 
stated that can be realized in trajectory analysis problems. 
simplicity, weighting is not considered; however, it would introduce no 
difficulty. 
For the sake of 
The vector equation Ax = b does not, in general, have a solution x. 
However, all candidates for a least squares solution (i.e., a solution vector 
x minimizing (Ax - b) (Ax - b)) must be solutions of the normal equations 
(ref. 6). 
3 
3 * 
A A x = A b  
8 
THEOREM IX. L e t  A be any matrix (mxn) an& b be any vector (mxl). 
* * 
A A x = A b  
The equation 
always has a solution, and hence a general solut ion i s  given by 
X = (A * A) + *  A b + [I - (A*A)+A*A) 
+ = A b + (I - A+A)y 
* 
Moreover, i f  A A i s  nonsingular, t he  solut ion i s  
+ x = A b  
and i s  unique. 
Proof: F i r s t ,  it i s  shown t h a t  
* * 
A A x = A b  
has a solution. Consider t h e  vector 
+ x = A b  
+ * * A A(A+b) = A b, Since equation (6) of THEOREM I11 implies 
a solut ion of equation (21). 
Corollary A, implies that t h e  general solut ion t o  equation (21) i s  
x = A b i s  indeed 
The existence of t h i s  solution, together with 
* + *  x = ( A  A )  A b +[I - (A*A)+A*A> 
By using equation (16) of THEOREM 111, 
+ x = A b  + (I - A+A)y 
* 
Finally,  if A A i s  nonsingular, then 
* + *  x = ( A  A) A b + (I - I ) y  
= A'b 
and equation (21) has a unique solution. 
I n  summary, if  x i s  a least squares solut ion of Ax = b, then x must 
s a t i s f y  
+k * 
A A x = A b  
9 
All solutions of t h i s  equation a r e  given by 
+ x = A b  + (I - A+A)y 
Any vector of t h e  form 
x = A'b + (I - A+A)y 
i s  a "candidate" for a l e a s t  squares solution, and t h i s  form describes the 
"class  of a l l  candidates . ' I  
Corollary D. Every solution of 
,e 5;- 
A A x = A b  
minimi z e s 
46 
Q = (AX - b)  (AX - b)  
provided Q has a m i n i m .  
Proof: Any vector a t  which Q i s  minimum i s  of the  form 
+ x = A b + (I - A+A)y 
If Q has a minimum, l e t  
+ + 
= A b + (I - A A ) y 2  x 1 
be any other solut ion.  I n  order t o  show t h a t  
Axl and Ax2 i s  examined i n  l i g h t  of THEOREM I. 
Axl = A[A+b + (I - A+A)yl] 
= AA'b + ( A  - AA+A)y, 
= AA'b + ( A  - A)yl  
= AA'b 
10 
Similarly, 
Ax2=AAb + 
SO that 
- b)*(Pxl - b) = <Ax2 - b)"(Ax2 - b) A"1 
that is, every vector of the form 
+ x = A b + (I - A+A)y 
yields the same minimum value of Q. 
SUBROUTINE GENINV 
GENINV is a FORTRAN IV subroutine used to compute the generalized inverse 
of an m n  matrix A .  All compuwtions are made in double-precision floating 
point arithmetic. 
explicit form . The subroutine 1 s  based on the algorithm suggested by the 
Calling Sequence 
Call GENINV (A, Ap, M, N, L, E) 
where : 
A 
AP 
M 
N 
L 
E 
double-dimensioned, double-precision array containing the original matrix. 
A is dimensioned A(25 ,  25) 
double-dimensioned, double-precision array where the generalized inverse 
of A will be computed. AP is dimensioned AP(25, 25) 
number of rows in the original matrix. 
number of columns in the original matrix 
twice N 
some small number for near-zero divisor test 
11 
Method 
Given A 
Compute : * 
C = A A  
2 c = cc 
(Print A) 
(Print C) 
(Print c ) 2 
Find nonsingular matrices E and P such that 
EC2P = (zr;) = I 0 
(Print E, P, Io) 
(A form of Gaussian elimination with pivoting employed) 
Compute : 
R = PIoE 
then 
c+ = "C
also 
+ + 36 A = C A  
Remarks 
(Print R) 
(Print c+> 
(Print A+> 
The program uses two double-precision arrays CSQ(30, 50)  and B(25, 
for internal manipulation. The subroutine leaves the original matrix 
Results are printed after each step as indicated. 
CONCLUSION 
25) 
intac ,. 
The theory developed in THEORFSIS VI1 and VI11 gives rise,to easy calcu- 
lation of the generalized inverse of an arbitrary complex matrix when only 
the solution to the matrix equation AXA = A can be found. In general, a 
12 
simultaneous solution must be found for four matrix equations, given in 
THEOREM I, that defjne the generalized inverse. 
Manned Spacecraft Center 
National Aeronautics and Space Administration 
Houston, Texas, February 5, 1965 
REFERENCES 
1. 
2. 
3 .  
4. 
5 -  
6 .  
7. 
8. 
10. 
Bjerhammar, A.: 
Squares With Special Reference to Geodetic Calculations. 
Inst. Tech., Stockholm, vol. 49, 1951, pp. 1-86. 
B u l l .  her. bkth. Soc., vol. 26, 1920, pp. 394-395. 
Application of Calculus of Matrices to Method of Least 
Trans. Roy. 
Moore, E. H.: 
Penrose, R.: A Generalized Inverse for mtrices. Proc. Camb. Philos. SOC., 
vol. 51, 1955, pp. 406413. 
Bjerhammar, A. : Rectangular Reciprocal &trices With Special Reference to 
Geodetic Calculations. Bull. Geodesique, 1.951, pp. 188-220. 
Decell, H.: An Explicit Form for the Generalized Inverse of an Arbitrary 
Complex mtrix. SIAM Rev., 1965. 
Greville, T. N. E.: The Pseudoinverse 01’ a Rectangular or Singular Iktrix 
and Its Applications to the Solution of Systems of Linear Equations. 
SIAM Rev., vol.  1, 1959, pp. 38-43. 
Penrose, R.: On Best Approximate Solution of Linear &trix Equations. 
Proc. Camb. Philos. SOC., vol.  52, 1956, pp. 17-19. 
Cecioni, F.: Sopra Operazioni Algebriche. Ann.  Scu. Norm. Sup. Pisa, 
voi. 11, igio, pp. 17-20. 
Cline, Randall E.: Note on the Generalized Inverse of the Product of 
bktrices. SLAM Rev., vol. 6, 1964, pp. 57-58. 
Cline, R. E.: Representations for the Generalized Inverse of mtrices 
With Applications in Linear Programming. Doctoral Thesis, Purdue 
University, 1963. 
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A special form of a generalized inverse of an arbitrary complex matrix

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