FACTORIZATION AND MOORE-PENROSE INVERSE

Gradimir V. Milovanovic´ and Predrag S. Stanimirovic´

Abstract. In this paper we develop a few representations of the Moore- Penrose inverse, based on full-rank factorizations of matrices. These rep- resentations we divide into the two different classes: methods which arise from the known block decompositions and determinantal representation. In particular cases we obtain several known results.

1. Introduction

Cm×n The set of m × n complex matrices of rank r is denoted by r = {X ∈ Cm×n |r : rank(X) = r}. With A and A|r we denote the first r columns of A and the first r rows of A, respectively. The identity of the order k is denoted by Ik, and O denotes the zero block of an appropriate dimensions. We use the following useful expression for the Moore-Penrose A†, based on the full-rank factorization A = PQ of A [1-2]: A† = Q†P † = Q∗(QQ∗)−1(P ∗P )−1P ∗ = Q∗(P ∗AQ∗)−1P ∗.

We restate main known block decompositions [7], [16-18]. For a given Cm×n matrix A ∈ r there exist regular matrices R, G, permutation matrices E, F and unitary matrices U, V , such that:

1991 Mathematics Subject Classification. 15A09, 15A24.

Typeset by AMS-TEX 1 2 G. V. Milovanovi´cand P. S. Stanimirovi´c

Ir O B O (T1) RAG = OO = N1, (T2) RAG = OO = N2,     I K I O (T ) RAF = r = N , (T ) EAG = r = N , 3 OO 3 4 K O 4     Ir O Ir O (T5) UAG = OO = N1, (T6) RAV = OO = N1,     B O B K (T7) UAV = OO = N2, (T8) UAF = OO = N5,     B O (T ) EAV = = N , 9 K O 6   A11 A12 (T10) EAF = = N7, where rank(A11)=rank(A). A21 A22   (T11) Transformation of similarity for square matrices [11]:

−1 ∗ −1 Ir K ∗ −1 T1 T2 RAR = RAF F R = OO F R = OO .     The block form (T10) can be expressed in two different ways:

A11 A11T (T10a) EAF = , where the multipliers S and T satisfy SA11 SA11T −1  −1  T = A11 A12, S = A21A11 (see [8]); A11 A12 A11 A12 (T10b) EAF = = −1 (see [9]). A21 A22 A21 A21A A12    11  Block representations of the Moore-Penrose inverse is investigated in [8], [11], [15–19]. In [16], [18] the results are obtained by solving the equations (1)–(4). In [15], [8] the corresponding representations are obtained using the block decompositions (T10a) and (T10b) and implied full-rank factorizations. Also, in [19] is introduced block representation of the Moore-Penrose in- verse, based on A† = A∗T A∗, where T ∈ A∗AA∗{1}.

Block decomposition (T11) is investigated in [11], but only for square matrices and the group inverse. The notion determinantal representation of the Moore-Penrose inverse of A means representation of elements of A† in terms of minors of A. Deter- minantal representation of the Moore-Penrose inverse is examined in [1–2], [4–6], [12–14]. For the sake of completeness, we restate here several nota- tions and the main result. For an m × n matrix A let α = {α1,... ,αr} and β = {β1,... ,βr} be subsets of {1,... ,m} and {1,... ,n}, respectively. RANK FACTORIZATION AND MOORE-PENROSE INVERSE 3

α ... α A 1 r Aα A Then β1 ... βr = | β | denotes the minor of determined by the rows α indexed by α and the columns indexed by β, and Aβ represents the corre- α sponding submatrix. Also, the algebraic complement of Aβ is defined by

∂ α ... α i α ... α α1 ... αp−1 αp+1 ... αr | Aα |=A 1 p−1 p+1 r =(−1)p+qA . β ij β1 ... βq 1 j βq+1 ... βr β1 ... βq−1 βq+1 ... βr ∂aij −     Adjoint matrix of a B is denoted by adj(B), and its deter- minant by |B|. Determinantal representation of full rank matrices is introuced in [1], and for full-rank matrices in [4–6]. In [12–14] is introduced an elegant derivation for determinantal representation of the Moore-Penrose inverse, using a full- rank factorization and known results for full-rank matrices. Main result of these papers is: Proposition 1.1. The (i, j)th element of the Moore-Penrose inverse G = Cm×n (gij ) of A ∈ r is given by

α ∂ α Aβ | Aβ | α:j∈α ; β:i∈β ∂aji gij = P γ γ . | A δ | | Aδ | γ,δ P In this paper we investigate two different representations of the Moore- Penrose inverse. The first class of representations is a continuation of the papers [8] and [15]. In other words, from the presented block factorizations of matrices find corresponding full-rank decompositions A = PQ, and then ap- ply A† = Q∗(P ∗AQ∗)−1P ∗. In the second representation, A† is represented in terms of minors of the matrix A. In this paper we describe an elegant proof of the well-known determinantal representation of the Moore-Penrose inverse. Main advantages of described block representations are their simply derivation and computation and possibility of natural generalization. Deter- minantal representation of the Moore-Penrose inverse can be implemented only for small dimensions of matrices (n ≤ 10).

2. Block representation Cm×n Theorem 2.1. The Moore-Penrose inverse of a given matrix A ∈ r can be represented as follows, where block representations (Gi) correspond to the block decompositions (Ti), i ∈ {1,... , 9, 10a, 10b, 11}: ∗ −1 ∗ † −1 ∗ −1|r −1 ∗ −1|r (G1) A = G |r R A G |r R ,        4 G. V. Milovanovi´cand P. S. Stanimirovi´c

∗ −1 ∗ † −1 ∗ −1|r −1 ∗ −1|r (G2) A = G |r R B A G |r R B ,

 ∗ −1  ∗  I  |r I  |r (G ) A† = F r R−1 AF r R−1 , 3 K∗ K∗       −1  † −1 ∗ ∗ −1 ∗ ∗ (G4) A = G |r [ Ir, K ] EA G |r [ Ir, K ] E,  −1  † −1 ∗ −1 ∗  (G5) A = G |r U|rA G |r U|r,  ∗ −1  ∗ † |r  −1|r |r  −1|r (G6) A = V R AV R ,

† |r ∗  |r −1  ∗   (G7) A = V B U|rAV B U|r, −1 B∗  B∗ (G ) A† = F U AF U , 8 K∗ |r K∗ |r     † |r ∗ ∗ |r −1 ∗ ∗ (G9) A = V [ B , K ] EAV [ B , K ] E, −1 Ir  Ir (G ) A† = F A∗ [ I , S∗ ] EAF A∗ [ I , S∗ ] E 10a T ∗ 11 r T ∗ 11 r     I = F r (I + T T ∗)−1A−1(I + S∗S)−1 [ I , S∗ ] E, T ∗ r 11 r r   1 A∗ A∗ − (G ) A† =F 11 (A∗ )−1[A∗ , A∗ ]EAF 11 (A∗ )−1[A∗ , A∗ ]E 10b A∗ 11 11 21 A∗ 11 11 21  12  12 A∗ =F 11 (A A∗ +A A∗ )−1A (A∗ A +A∗ A )−1[A∗ , A∗ ]E, A∗ 11 11 12 12 11 11 11 21 21 11 21  12 ∗ −1 ∗ † ∗ Ir −1|r ∗ Ir −1|r (G11) A =R −1 ∗ R T1 AR −1 ∗ R T1 . T1 T2 T1 T2          Proof. (G1) Starting from (T1), we obtain O −1 Ir −1 −1 Ir O −1 A = R OO G = R O [ Ir, ] G ,     which implies

−1 Ir −1|r O −1 −1 P = R O = R , Q = [ Ir, ] G = G |r.   Now, we get ∗ −1 ∗ † ∗ ∗ ∗ −1 ∗ −1 ∗ −1|r −1 ∗ −1|r A = Q (P AQ ) P = G |r R A G |r R .        RANK FACTORIZATION AND MOORE-PENROSE INVERSE 5

The other block decompositions can be obtained in a similar way.

(G5) Block decomposition (T5) implies O ∗ Ir −1 ∗ Ir O −1 A = U OO G = U O [ Ir, ] G ,     which means

∗ Ir ∗|r O −1 −1 P = U O = U , Q = [ Ir, ] G = G |r.  

(G7) It is easy to see that (T7) implies O ∗ B ∗ ∗ B O ∗ A = U OO V = U O [ Ir, ] V .     Thus,

∗ B ∗|r O ∗ ∗ P = U O = U B, Q = [ Ir, ] V = V |r,   which means ∗ ∗ ∗ |r P = B U|r, Q = V .

(G10a) From (T10a) we obtain

∗ A11 A11T ∗ ∗ Ir ∗ A = E F = E A11 [ Ir, T ] F , SA11 SA11T S     which implies, for example, the following full rank factorization of A: I P = E∗ r A , Q = [ I , T ] F ∗. S 11 r   Now, 1 I , I , − A† = F r A∗ [ I , S∗ ] EAF r A∗ [ I , S∗ ] E. T ∗ 11 r T ∗ 11 r     The proof can be completed using I EAF = r A [ I , T ] .  S 11 r  

−1 Remarks 2.1. (i) A convenient method for finding the matrices S, T and A11 , required in (T10a) was introduced in [8], and it was based on the following extended Gauss-Jordan transformation: 6 G. V. Milovanovi´cand P. S. Stanimirovi´c

A A I I T A−1 11 12 → 11 . A21 A22 O OO −S     (ii) In [3] it was used the following full-rank factorization of A, derived A11 A12 from A = −1 : A21 A21A A12  11  A11 −1 P = , Q = [ Ir, A11 A12 ] . A21   3. Determinantal representation

In the following definition are generalized concepts of determinant, alge- braic complement, adjoint matrix and determinantal representation of gen- eralized inverses. (see also [14].) Definition 3.1. Let A be m × n matrix of rank r.

(i) The generalized determinant of A, denoted by Nr(A), is equal to α α Nr(A)= Aβ | Aβ | , α,β P (ii) Generalized algebraic complement of A corresponding to aij is

† α ∂ α Aij = Aβ | Aβ | . α:j∈α;β:i∈β ∂aji P † (iii) Generalized adjoint matrix of A, denoted by adj (A) is the matrix † whose elements are Aij. For full-rank matrix A the following results can be proved: Lemma 3.1. [14] If A is an m × n matrix of full-rank, then: | AA∗ | , r = m (i) Nr(A)= | R∗A | , r = n.  (A∗ adj(AA∗)) , r = m A† ij (ii) ij = ∗ ∗ ( ( adj(A A)A )ij , r = n. A∗(AA∗)−1, r = m (iii) A† = (A∗A)−1A∗, r = n.  A∗ adj(AA∗), r = m (iv) adj†(A)= adj(A∗A)A∗, r = n.  Main properties of the generalized adjoint matrix, generalized algebraic complement and generalized determinant are investigated in [14]. RANK FACTORIZATION AND MOORE-PENROSE INVERSE 7

Lemma 3.2. [14] If A = PQ is a full-rank factorization of an m×n matrix A of rank r, then (i) adj†(Q) · adj†(P )= adj†(A);

(ii) Nr(Q) · Nr(P )= Nr(P ) · Nr(Q)= Nr(A);

From Lemma 3.1 and Lemma 3.2 we obtain an elegant proof for the determinantal representation of the Moore-Penrose inverse. Theorem 3.1. Let A be an m×n matrix of rank r, and A = PQ be its full- rank factorization. The Moore-Penrose inverse of A posseses the followig determinantal representation:

α ∂ α Aβ | Aβ | † α:j∈α ; β:i∈β ∂aji aij = P γ γ . | A δ | | Aδ | γ,δ P Proof. Using A† = Q†P † [3] and the results of Lemma 3.1 and Lemma 3.2, we obtain: Q∗ adj(QQ∗) adj(P ∗P )P ∗ A† = Q†P † = Q∗(QQ∗)−1(P ∗P )−1P ∗ = = | QQ∗ | | P ∗P | adj†(Q)adj†(P ) adj†(A) = = .  Nr(Q)Nr(P ) Nr(A)

4. Examples

Example 4.1. Block decomposition (T1) can be obtained by applying trans- formation (T3) two times:

Ir K R1AF1 = OO = N3,   O T Ir R2N3 F2 = OO = N1.   Then, the regular matrices R, G can be computed as follows:

T T T T T T T N1 = N1 = F2 N3R2 = F2 R1AF1R2 ⇒ R = F2 R1, G = F1R2 . −11 3 5 7 For the matrix A = 1 0 −2 0 4 we obtain  1 1 −1 5 15  −1 2 4 10 18   8 G. V. Milovanovi´cand P. S. Stanimirovi´c

0 100 1 100 R1 = , F1 = I5,  −1 −2 1 0  −2 −1 0 1   1 0000 0 1000 R2 =  2 −1 1 0 0  , F2 = I4. 0 −5 0 1 0  −4 −11 0 0 1    T From R = R1, G = R2 , we get −1 1 |2 R−1 = 1 0 , G−1 = 1 0 −2 0 4 .  1 1  |2 0 1 1 5 11 −1 2     Using formula (G1), we obtain 169 67 233 137 − 6720 2240 6720 − 6720 1 1 1 1  128 − 128 − 128 128  A† =  781 − 330 − 1037 653  .  13440 4480 13440 13440     5 − 5 − 5 5   128 128 128 128     − 197 151 709 59   13440 4480 13440 13440    −1 0 1 2 −1 1 0 −1 Example 4.2. A 0 −1 1 3 For the matrix =  0 1 −1 −3  we obtain  1 −1 0 1   1 0 −1 −2    1 −1 1 −1 0 −1 −2 A− = , S = −1 1 , T = [5]. 11 −1 1  0 −1  −1 −3   −1 0     Using (G10a) we obtain 5 3 1 1 3 5 − 34 − 17 34 − 34 17 34 4 13 5 5 13 4  51 102 − 102 102 − 102 − 51  A† = .  7 5 1 − 1 − 5 − 7   102 102 51 51 102 102     1 − 1 3 − 3 1 − 1   17 34 34 34 34 17    RANK FACTORIZATION AND MOORE-PENROSE INVERSE 9

4 −1 1 2 Example 4.3. For the matrix A = −2 2 0 −1 we obtain  6 −3 1 3  −10 4 −2 −5 −1 000   −1 100 1 0 −1 −2 R = , T1 = , T2 = , F = I4.  −1 110  0 1 −1 −3 2 −1 0 1       Then, the following results can be obtained:

∗ −4 −6 −1|r −1 −1 0 1 ∗ Ir 1 3 R T1 = ,R −1 ∗ = . 0 1 −1 1 T1 T2  −1 −1        −2 −3    Finally, using (G11), we get 8 10 2 2 81 81 − 81 − 27 47 79 16 5  162 162 − 81 − 54  A† = .  7 11 − 2 − 1   54 54 27 18     4 5 − 1 − 1   81 81 81 27    References

[1] Arghiriade, E. e Dragomir, A., Une nouvelle definition de l’inverse generalisee d’une matrice, Rendiconti dei Lincei, serir XIII Sc. fis. mat. e nat. 35 (1963), 158–165. [2] Bapat, R.B.; Bhaskara, K.P.S. and Manjunatha Prasad, K., Generalized inverses over integral domains, Linear Algebra Appl. 140 (1990), 181–196. [3] Ben-Israel, A. and Grevile, T.N.E., Generalized inverses: Theory and applications, Wiley-Interscience, New York, 1974. [4] Gabriel, R., Das Verallgemeinerte Inverse einer Matrix u¨ ber einem beliebigen K¨or- per - analytish betrachtet, J. Rewie Ansew Math. 244(V) (1968), 83–93. [5] Gabriel, R., Extinderea complementilor algebrici generalizati la matrici oarecare, Studii si Cercetari Matematice 17 -Nr. 10 (1965), 1566–1581. [6] Gabriel, R., Das allgemeine element des Verallgemeinerte Inverse von Moore-Pen- rose, Rev. Roum. Math. Pures Appl. 37 No 6 (1982), 689–702. [7] Lancaster, P. and Tismenetsky, M., The Theory of Matrices with Applications, Academic Press, 1985. [8] Noble, P., A methods for computing the generalized inverse of a matrix, SIAM J. Numer. Anal. 3 (1966), 582–584. [9] Penrose, R., A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406–413. [10] Penrose R., On best approximate solution of linear matrix equations, Proc. Cam- bridge Philos. Soc. 52 (1956), 17–19. 10 G. V. Milovanovi´cand P. S. Stanimirovi´c

[11] Robert, P., On the Group inverse of a linear transformation, J. Math. Anal. Appl. 22 (1968), 658–669. [12] Stanimirovi´c, P. and Stankovi´c, M., Generalized algebraic complement and Moore- Penrose inverse, Filomat 8 (1994), 57–64. [13] Stanimirovi´c, P. and Stankovi´c, M., Determinantal representation of weighted Moo- re-Penrose inverse, Matematiˇcki Vesnik 46 (1994), 41–50. [14] Stanimirovi´c, P. and Stankovi´c, M., Determinantal representation of generalized inverses over integral domains, Filomat 9:3 (1995), 931–940. [15] Tewarson, R.P., A direct method for generalized matrix inversion, SIAM J. Numer. Anal. 4 No 4 (1967), 499–507. [16] Zielke, G., A survey of generalized matrix inverse, Banach Center Publication 13 (1984), 499–526. [17] Zielke, G., Die Aufl¨osung beliebiger linearer algebraisher Gleichungssysteme durch Blockzerlegung, Beitr¨age zur Numerischen Mathematik 8 (1980), 181–199. [18] Zielke, G., Motivation und Darstellung von verallgemeinerten Matrixinversen, Bei- tr¨age zur Numerischen Mathematik 7 (1979), 177–218. [19] Zlobec, S., An explicit form of the Moore-Penrose inverse of an arbitrary complex matrix, SIAM Rev. 12 (1970), 132–134.

Department of Mathematics, Faculty of Electronic Engineering, P.O.Box 73, Beogradska 14, 18000 Niˇs

University of Niˇs, Faculty of Philosophy, Departement of Mathematics, Cirila´ i Metodija 2, 18000 Niˇs