INVERSES OF SQUARE MATRICES AND RATIONAL ...

{1} -INVERSES OF SQUARE MATRICES

AND RATIONAL CANONICAL FORM Ljubiˇ sa Koˇ cinac, Predrag Stanimirovi´ c and Slaviˇ sa Djordjevi´ c Faculty of Philosophy, Department of Mathematics, Niˇs

Abstract.

In this paper we solve the first Penrose’s equation AXA = A for square real matrices A using the rational canonical form of matrices. The idea is to find {1}- inverse X of A using similarity X = T ZT −1 , where Z is {1}-inverse of B and A = T BT −1 is the rational canonical representation of A.

1. Introduction Let Rn×n be the set of n × n real matrices and let O denote the zero matrix of an appropriate size. For an m × n complex matrix A there exists a unique n × m matrix A† = X which satisfies the following equations, introduced by R. Penrose [8] and called the Penrose’s equations. (1)

AXA = A

(2) XAX = X

(3) (AX)T = AX

(4) (XA)T = XA,

where AT denotes conjugate and transpose of matrix A. The matrix X is known as the Moore-Penrose generalized inverse of A. For a sequence S of the set {1, 2, 3, 4} the set of matrices G satisfying the conditions contained in S will be denoted by A{S}. A matrix G in A{S} is called an S-inverse of A and is denoted by A(S) . For the sake of completeness we present a brief description of the rational canonical form [5], [10]. Definition 1.1. Given a monic polynomial (1.1)

p(t) = tk + ak−1 tk−1 + . . . + a1 t + a0 .

The matrix AMS Subject Classification (1991): 15A09, 15A24 1

2



(1.2)

0  1  A= 0  ... 0

0 0 1 ... 0

... ... ... ... ...

0 0 0 ... 1

 −a0 −a1   −a2  ∈ Rk×k  ... −ak−1

is known as the companion matrix of the polynomial (1.1). Theorem 1.1. Every matrix A ∈ Rn×n is similar to the direct sum of companion matrices of its (real) elementary divisors. Consequently, if A = T BT −1 is the rational canonical representation of A ∈ Rn×n then the block diagonal matrix B can be represented in the form   B1 . . . O O ... O ... ... ...  ... ... ...   O ... O   O . . . Bq (1.3) B =   = B1 ⊕ · · · ⊕ Bq ⊕ Bq+1 ⊕ · · · ⊕ Bp ,  O . . . O Bq+1 . . . O    ... ... ... ... ... ... O ... O O . . . Bp where the blocks Bi , 1 ≤ i ≤ p are the companion matrices of elementary divisors tmi + aimi −1 tmi −1 + . . . + ai1 t + ai0 , and have the form

(1.4)



0  1  Bi =  0  ... 0

0 0 1 ... 0

... ... ... ... ...

0 0 0 ... 1

 −ai0 −ai1   −ai2  ∈ Rmi ×mi .  ... −aimi −1

We can suppose, without loss of generality, that the blocks B1 , . . . , Bq are invertible, i.e. ai0 ̸= 0, i = 1, . . . q and Bq+1 , . . . , Bp are noninvertible, i.e. ai0 = 0, i = q + 1, . . . p. In the second section we obtain representation of {1}-inverses of real square matrices in terms of blocks of the rational canonical form. In this way we obtain an effective block representation of {1}-inverses. More precisely, we solve the Penrose’s equation AXA = A using the method presented in [4], [9] and the rational canonical form instead of the Jordan canonical form. Such an approach has not been, as far as we know, employed before. In third section of the paper we develop several examples.

2. Representation of {1}-inverses Let A be a given real matrix of the order n, let B = T −1 AT be its rational canonical form and X be an arbitrary {1}-inverse of A. Using elementary fact: X = A(1) = T B (1) T −1 , we first compute an arbitrary {1}-inverse Z of B, and then obtain X = T ZT −1 . For this

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( ) i=1,... ,p purpose, we find a partition Zij , j=1,... ,p of the matrix Z = T −1 XT , where the blocks Zij are of the size mi × mj , convenient to the partition of the given B. First we investigate relations between the blocks Bγ , γ = 1, . . . , p and the blocks Zij . Lemma 2.1. Let A ∈ Rn×n and let A = T BT −1( be its rational canonical representation, ) i=1,... ,p −1 X ∈ A{1} and Z = T XT . Then the blocks Zij , j=1,... ,p and the blocks Bγ , γ = 1, . . . , p satisfy the following conditions: (B1)

Zii = Bi−1 ,

(B2)

Zij = O,

(B3)

Bi Zii Bi = Bi ,

(B4)

Zij Bj = O,

(B5)

Bi Zij = O,

(B6)

Bi Zij Bj = O,

i = 1, . . . , q ( i=1,... ,q ) j=1,... ,q

(2.1)

(

,

i ̸= j

i = q + 1, . . . , p i=1,... ,q j=q+1,... ,p

)

( i=q+1,... ,p ) j=1,... ,q

( i=q+1,... ,p ) j=q+1,... ,p

.

where Bi , i = 1, . . . , p are square matrices of the order mi and Zij ∈ Rmi ×mj . Proof. It is easy to verify that the equation AXA = A is equivalent with BZB = B. Now, from the equation BZB = B we get { Bi , i = j (2.2) Bi Zij Bj = O, i ̸= j. Then (B1)-(B6) follow from invertibility of the blocks Bi , i = 1, . . . , q.



We are now ready for the main theorem, in which the general form of the class of {1}inverses for the rational canonical form is obtained. Theorem 2.1. Let A = T BT −1 be the rational canonical representation of A ∈ Rn×n and let X be an arbitrary {1}-inverse of A. Then X = T ZT −1 , where Z ∈ B{1}. Moreover, Z has the following block representation:  −1  B1 ... O Z1,q+1 ... Z1,p ... ... ... ... ...   ...   ... Bq−1 Zq,q+1 ... Zq,p   O (2.3) Z= .  Zq+1,1 . . . Zq+1,q Zq+1,q+1 . . . Zq+1,p    ... ... ... ... ... ... Zp,1 ... Zp,q Zp,q+1 ... Zp,p

4 ij If zα,β represents (αβ)-element of the block Zij , the blocks Zij are as follows:   ij z11 0 ... 0  z ij ( i=1,... ,q ) 0 ... 0  21 , (Z1) Zij =  j=q+1,... ,p ;  ...  ... ... ... ij zm 0 ... 0 i ,1



(Z2)

ij ai1 zm i ,1  ij  ai2 zmi ,1  ... Zij =   i ij  ami −1 zmi ,1 ij zm i ,1

 (Z3)

ij z11

 ij  z21  Zij =  . . .  ij  zmi −1,1 ij zm i ,1 

(Z4)

ii z11 ii z21 ii z31 ...

   Zii =     z ii

mi −1,1 ii zm i ,1

ij ai1 zm i ,2 ij ai2 zm i ,2 ... ij aimi −1 zm i ,2 ij zmi ,2

... ... ... ... ...

 ij ai1 zm i ,mj  ij a2i zm  i ,mj  ... ,  i ij ami −1 zmi ,mj  ij zm i ,mj

ij ai1 zm i ,2

...

ij ai1 zm i ,mj

ij z1,m j

ij ai2 zm i ,2 ... ij aimi −1 zm i ,2 ij zmi ,2

... ... ... ...

ij ai2 zm i ,mj ... ij aimi −1 zm i ,mj ij zmi ,mj −1

ii 1 + ai1 zm i ,2 ii ai2 zm i ,2 ij ai3 zm i ,2 ... ii aimi −1 zm i ,2 ii zm ,2 i

... ... ... ... ... ...

 ii ai1 zm i ,mi ij  ai2 zm i ,mi  i ij  a3 zmi ,mi ,  ...  ii i 1 + ami −1 zmi ,mi  ii zm i ,mi

( i=q+1,... ,p ) j=1,... ,q

;



 ij z2,m  j  ... ,  ij zm  i ,mj ij zmi ,mj

(

i=q+1,... ,p j=q+1,... ,p i̸=j

) ;

( i=q+1,... ,p ) .

Proof. General form of Z can be obtained solving the equations (B1)-(B6). From (B4) we get the following system of linear equations: ( )  ij α=1,... ,mi z = 0, ,  β=2,... ,m j  α,β (2.4) m∑ j −1  ij  ajl zc,l+1 = 0, (c = 1, . . . , mi ) . l=1

It is easy to verify that the second equation in (2.4) is implied by the first, and we obtain (Z1). Similarly, starting from (B5) we obtain ij ij zα,β − aiα zm = 0, i ,β

which implies (Z2).

(

α=1,... ,mi −1 β=1,... ,mj

) ,

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Now, starting from (B6) we obtain ( )  ij p=1,... ,mi −1 i ij z − a z = 0, ,  q=2,... ,mj p mi ,q  p,q (2.5) ( ) m∑ j −1  ij ij  ajl zc,l+1 − aic zm = 0, (c = 1, . . . , mi − 1). ,l+1 i l=1

The second system is a consequence of the first, which implies (Z3). Finally, equation (B3) implies the following system of linear equations:  ii ii − aip zm = 1, (p = 1, . . . , mi − 1) , z  i ,p+1  p,p+1   m∑ ( ) i −1 ii ii ail zc,l+1 − aic zm = aic , (c = 1, . . . , mi − 1) , (2.6) i ,l+1  l=1   ( p=1,... ,mi −1 )  ii ii zp,q − aip zm = 0, . ,q q=2,... ,mi i The second equation in (2.6) is implied by the first and third equation in (2.6), and consequently { ii 1 + aip zm , q =p+1 i ,q ii zp,q = ( p=1,... ,mi −1 ) i ii ap zmi ,q , q ̸= p + 1, , q=2,... ,mi 

which is equivalent with (Z4).

Remark 2.1. The class of {1, 2, 3} and {1, 2, 4}-inverses of an arbitrary matrix A can be computed using the class of {1}-inverses of the matrix AAT , according to the following theorem [1]: for every matrix A with complex elements AT (AAT )(1) ∈ A{1, 2, 4} and (AT A)(1) AT ∈ A{1, 2, 3}. We can use the following fact, useful for construction of the rational canonical form of AAT [7]: if A is the companion matrix of the polynomial p(t) = tk + ak−1 tk−1 + . . . + a1 t + a0 , then the characteristic polynomial of AAT is ( ( ) ) k ∑ 2 2 k−2 2 (t − 1) t − 1+ | ai | t + | ak | = 0. i=0

3. Examples 

 0 2 0 −6 2 0 2  1 −2 0   Example 3.1. For the matrix A =  1 0 1 −3 2  we obtain the following rational   1 −2 1 −1 2 1 −4 3 −3 4 canonical representation: A = T BT −1 , where [10]     1 0 0 2 0 0 −2 0 0 0  1 0 0 0 0   0 1 1 −2 1      B =  0 0 0 −2 0  and T =  0 1 0 0 1.  0 0 1 0 0    0 1 0 −2 1 0 0 0 0 2 0 1 0 −4 2

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From Theorem 2.1. we obtain 

B1−1 Z= O O



O B2−1 O

0 1 0   − 12 0 0  O  0 0 0 O =  − 12 −1 0 0  B3  0 0 0

and



X = A(1) = A−1 = T ZT −1

0

  1  −2    =  − 12    1  −2  − 12

0 0 1 0

 0 0    0  , 0  

0

1 2

−1

0

3

−1

1

3 2

−3

1 2

0

1 − 32

1

1 − 52

2

3 2

− 92



     1  . 2    1  2   1



 0 0 0 0 −2 3  0 0 0 0   Example 3.2. Consider A =  0 0 0 0 1 . Rank of this matrix is 2, so 3   0 0 0 0 4 −2 3 1 4 1 eigenvalues of A are equal to zero. For the rest eigenvalues λ1 and λ2 the following relations are valid [7]: ( ) λ1 + λ2 = 1, λ1 · λ2 = − (−2)2 + 32 + 12 + 42 = −30. It is easy to verify λ1 = −5, λ2 = 6, the characteristic polynomial of A is λ3 (λ + 5) (λ − 6).   −5 0 0 0 0  0 6 0 0 0      The rational canonical form of A is B =  0 0 0 0 0 , and transformation of    0 0 0 0 0  0 0 0 0 0   2 2 1 0 0  −3 −3 0 1 0    similarity is given by the matrix T =  −1 −1 2 −3 −4  .   −4 −4 0 0 1 5 −6 0 0 0 According to Theorem 2.1. we get the following representation for Z = T −1 XT :

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 

B1−1  0  Z =  Z31  Z41 Z51  −1 5

 0  =  z31  z41 z51

0 B2−1 Z32 Z42 Z52 0 1 6 z32

z42 z52

Z13 Z23 Z33 Z43 Z53 z13 z23 z33 z43 z53

Z14 Z24 Z34 Z44 Z54

z14 z24 z34 z44 z54

− 15

0

13 z11

14 z11

15 z11

  Z15  23 24 25  z11 z11 0 16 z11 Z25     31 32 33 34 35 Z35  =  z11 z11 z11 z11 z11   Z45  41 42 43 44 45  z11 z11 z11 z11 z11 Z55  51 52 53 54 55 z11 z11 z11 z11 z11 

      =    

z15 z25   z35  , zij arbitrary .  z45 z55

Finally, an arbitrary {1}-inverse X of A is equal to X = T ZT −1 . 

2 1 −1 2  −5 −3 Example 3.3. Let A =  2 2 0 −3 −1 2 mial of A are λ3 (λ − 3).

 −4 9 . The characteristic and minimal polyno−5 4

 3 0 0 0   Therefore, the rational canonical form of A is B =  0 0 0 0  , and transformation 0 1 0 0 0 0 1 0   −1 1 1 1  2 0 −3 −1 of similarity is given by T =   . According to Theorem 2.1. we get the −1 1 2 1 1 0 −1 0 following representation for the class of {1}-inverses of B:  1  12   1 0 0 z 11 z12 0 0 3 )  3 ( −1  0  B1 Z12   0 z22 1 22 1 0  Z= =  0 z11 = , 0 z 0 1 Z21 Z22  0 z 22  32 0 1 21 z41 z42 z43 z44 z 21 z 22 z 22 z 22 31

31



32

33

where zij are arbitrary. Finally the class A{1} can be generated from the equation X = T ZT −1 . REFERENCES [1]

Ben-Israel, A. ; Grevile, T.N.E., Generalized Inverses: Theory and Applications, Wiley-Interscience, New York, 1974.

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[2] [3] [4] [5]

Campbell, S.L. ; Meyer, C.D., Generalized Inverses of Linear Transformations, Pitman, New York, 1979. Erdelyi, I., On the matrix equation Ax = λBx, J. Math. Analysis Appl. 17 (1967), 119–132. Giurescu, C., Gabriel, R., Unele proprietati ale matricilor inverse generalizate si Semiinverse, An. Univ. Timisoara, Ser. Sci Mat.-Fiz. 2 (1964), 103–111. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge Univ. Press, Cambridge, 1985.

[6]

Koˇcinac, Lj., Linear Algebra and Analytic Geometry, University of Niˇs, 1991. (In Serbian).

[7]

Mitrinovi´c, D.S., Matrices and Determinants, Nauˇcna Knjiga, Beograd, 1975 (In Serbian).

[8]

Penrose, R., A generalized inverse for matrices, Proc. Cambridge Phil. Soc. 51 (1955), 406–413.

[9]

Stanimirovi´c, P., Moore-Penrose and group inverse of square matrices and Jordan canonical form, Circolo Matematico di Palermo 44 (1995). [10] Stephen, F.H. and Arnold, I.J., Linear Algebra, Prentice-Hall International, Inc., 1989. University of Niˇs, Faculty of Philosophy, Department of Mathematics, ´ Cirila i Metodija 2, 18000 Niˇs, Yugoslavia