Computing {2,4} and {2,3}-inverses Using SVD-like Factorizations and QR Factorization

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1 Filomat 30:2 (2016) DOI /FIL S Publihed by Faculty of Science and Mathematic Univerity of Niš Serbia Available at: Computing {24} and {23}-invere Uing SVD-like Factorization and QR Factorization Bilall I Shaini a a State Univerity of Tetova Rr e Ilindenit pn Tetovo R Macedonia Abtract We derive condition for the exitence and invetigate repreentation of {2 4} and {2 3}- invere with precribed range T and null pace S A general computational algorithm for {2 4} and {2 3} generalized invere with given rank and precribed range and null pace i derived The algorithm i derived generating the full-rank repreentation of thee generalized invere by mean of variou complete orthogonal matrix factorization More preciely computational algorithm for {2 4} and {2 3}-invere of a given matrix A i defined uing an unique approach on SVD QR and URV matrix decompoition of appropriately elected matrix W 1 Introduction and Preliminarie Let C m n and Cr m n denote the et of all complex m n matrice and all complex m n matrice of rank r repectively I denote the unit matrix of an appropriate order By A R(A) rank(a) and N(A) we denote the conjugate tranpoe the range the rank and the null pace of A C m n The problem of calculating generalized peudoinvere i narrowly related with the four Penroe equation (1) AXA = A (2) XAX = X (3) (AX) = AX (4) (XA) = XA It i uual convention to denote by A{S} the et of all matrice obeying the condition contained in S In thi regard any matrix from A{S} i called S-invere of A and it i denoted by A (S) By A{S} we denote the et of all S-invere of A with rank The Moore-Penroe invere of A i a ingle element in the et A{ } and it i denoted by A For other important propertie of generalized invere ee [1 21 The et of outer (or {2}-invere) with precribed range and null pace i very important and frequently invetigated If A Cr m n T i a ubpace of C n of dimenion r and S i a ubpace of C m of dimenion m then A ha a {2}-invere X uch that R(X) = T and N(X) = S if and only if AT S = C m in which cae X i unique and we denote it by A (2) TS Full-rank repreentation of {2}-invere with precribed range and null pace i determined in the next propoition which i originated in [ Mathematic Subject Claification Primary 15A09 Keyword Generalized invere A (2) ; {23}-invere; {24}-invere;SVD decompoition; QR decompoition; Full-rank repreentation TS Received: 10 March 2014; Accepted: 09 June 2014 Communicated by Predrag Stanimirović addre: bilallhaini@uniteedumk (Bilall I Shaini)

2 B I Shaini / Filomat 30:2 (2016) Propoition 11 [12 Let A C m n r T be a ubpace of C n of dimenion r and let S be a ubpace of C m of dimenion m In addition uppoe that W C n m atifie R(W) = T N(W) = S Let W ha an arbitrary full-rank decompoition that i W = FG If A ha a {2}-invere A (2) TS then: (1) GAF i an invertible matrix; (2) A (2) TS = F(GAF) 1 G There exit many full-rank repreentation for different generalized invere of precribed rank [ A generalization of known repreentation for {2 3} {2 4}-invere of precribed rank from [1 i introduced in [20 In Propoition 12 we reduce known reult from [20 to contant complex matrice Propoition 12 [20 Let A Cr m n and 0 < r m 1 n 1 be choen integer Then the following general repreentation for {2 4} and {2 3}-invere of precribed rank are valid: (a) A{2 4} = { } (GA) G G C n1 m GA C n 1 n (b) A{2 3} = { F(AF) F C n m 1 AF C m m 1 } Propoition 13 from [16 exactly ditinguihe et A{2 4} and A{2 3} a two proper ubet of the et A{2} Propoition 13 [16 Let A C m n r be the given matrix and 0 < r a choen integer Aume that G C m and F C n are two arbitrary matrice atifying rank(ga) = rank(g) and rank(af) = rank(f) Then the following tatement are valid: (a) (b) A{2 4} = { (GA) (GA(GA) ) 1 G G C m A{2 3} = { F ((AF) AF) 1 (AF) F C n } GA C n ; } AF C m ; The Moore-Penroe invere A The weighted Moore-Penroe invere A MN The Drazin invere AD and the group invere A # are particular appearance of the generalized invere A (2) for appropriate choice of TS the matrix W which i exploited in Propoition 11 More preciely thee peudo-invere can be derived in particular cae W = A W = A = N 1 A M W = A l l ind(a) and W = A repectively (ee for example [1) In [16 we invetigate repreentation and computation of {2 4} and {2 3}-invere with precribed range and null pace Propoition 14 [16 For arbitrary matrix A C m n r and arbitrary integer atifying 0 < r we have (a) A{2 4} = { A (24) N(GA) N(G) G C m rank(ga) = rank(g) } (11) (b) A{2 3} = { A (23) R(F)R(AF) F C n rank(af) = rank(f) } (12) Variou repreentation of {2 3} and {2 4}-invere with precribed range and null pace ha been invetigated [ The expreion for {2 3} and {2 4}-invere of a normal matrix by it Schur decompoition are dicued in [26 But thee repreentation are not exploited in developing of ome effective computational procedure Additionally the general repreentation of {2 4} and {2 3}- invere of the form (GA) G and F(AF) repectively are not widely exploited in the literature we can emphaize the effective numerical method from [13 which are baed on everal modification of the hyper-power method Effective full-rank repreentation of the et A{2 4} and A{2 3} a particular cae of the full-rank repreentation of the et A{2} are derived in [16 Introduced full-rank repreentation enable adaptation of well-known algorithm for computing outer invere with precribed range and null pace into correponding algorithm for computing {2 4} and {2 3}-invere Correponding adaptation of the ucceive matrix quaring algorithm from [15 i developed in the paper [16 In the preent paper we continue and expand the reult from [18 19

3 B I Shaini / Filomat 30:2 (2016) The reult derived in the preent paper can be divided in two part Firtly we derive everal additional reult concerning repreentation of {2 4} and {2 3}-invere Later uing thee repreentation we derive numerical algorithm for computing {2 4} and {2 3}-invere More preciely we oberve that derived repreentation are matrix expreion (GA) G F(AF) involving generalized invere of a matrix product For thi purpoe we define and exploit SVD QR and URV matrix decompoition adopted for the matrix product and later compute it generalized invere In thi way we define algorithm for computing {2 4} and {2 3}-invere baed on thee SVD QR and URV decompoition The paper i organized a follow Neceary and ufficient condition which enure that an arbitrary {2 4} and {2 3}-invere repreent an invere with precribed range T and null pace S are conidered in the econd ection Variou repreentation of {2 4} and {2 3}-invere ariing from matrix decompoition are introduced and invetigated in the third ection Algorithm ariing from defined repreentation i given in Section 4 Numerical example are preented in the lat ection 2 Repreentation of {24} and {23}-invere with Precribed Range and Null Space The next tatement from [1 are ued a auxiliary reult Propoition 21 1 If the matrix F in the matrix product A = FBG i of full-column rank and G i of full-row rank then rank(a) = rank(b) 2 R(AB) = R(A) if and only if rank(ab) = rank(a) and N(AB) = N(B) if and only if rank(ab) = rank(b) 3 A = (A A) A = A (AA ) 4 N(A) = R(A ) N(A ) = R(A) In Lemma 21 we derive condition for the exitence and repreentation of {2 4} and {2 3}-invere which continue correponding reult from [3 Lemma 33 Lemma 21 Let A Cr m n be given m n matrix of rank r and 0 < r i elected integer Aume that T i a ubpace of C n of dimenion r and S i a ubpace of C m of dimenion m Let G C m be an arbitrary matrix atifying N(G) = S and rank(ga) = rank(g) = Then {2 4}-invere X := (GA) G atifie X = A (24) TS with R(X) = T N(X) = S if and only if N(GA) = T or T = R(A G ) Proof According to Propoition 13 and Propoition 11 we get (GA) G = (GA) (GA(GA) ) 1 G = A (24) N(GA) N(G) Now we conclude X := A (24) N(GA) N(G) = A(24) TS if and only if N(GA) = T or N(GA) = T The proof can be completed uing part 4 from Propoition 21 The proof of dual Lemma 22 i imilar and will be omitted Lemma 22 Let A C m n r be given m n matrix of rank r and 0 < r i elected integer Aume that T i a ubpace of C n of dimenion r and S i a ubpace of C m of dimenion m Let F C n be an arbitrary matrix atifying R(F) = T and rank(af) = rank(f) = Then {2 3}-invere X := F(AF) atifie X = A (23) TS with R(X) = T N(X) = S if and only if R(AF) = S or S = N(F A ) Comparing Propoition 11 with Propoition 13 we conclude : - {2 4}-invere are ubet of outer invere generated by the choice W = (GA) G ie F = (GA) - {2 3}-invere are ubet of outer invere generated by the choice W = F(AF) ie G = (AF) In the next tatement it i hown that {2 4}-invere can be generated uing only the matrix G and {2 3}-invere can be generated uing only the matrix F

4 B I Shaini / Filomat 30:2 (2016) Lemma 23 Let A C m n r and W C n m atifie R(W) = T N(W) = S (a) In the cae W = (GA) G we have (WA) W = (GA) G (b) In the cae W = F(AF) we have W(AW) = F(AF) Proof The proof can be derived uing the property 3 of the Moore-Penroe invere from Propoition 21 3 Computing {24} and {23}-invere by Matrix Factorization In thi ection we derive numerical method for computation of {2 4} and {2 3}-invere uing their general repreentation from the previou ection and variou matrix factorization A complete orthogonal factorization of an m n matrix A of rank r i any factorization of the form A = U [ T where T i r r quare noningular matrix [9 In the preent paper we exploit three appearance of the complete orthogonal factorization: the ingular value decompoition (SVD) QR decompoition (QRD) and the URV decompoition (URVD) Let A C m n r be given and 0 < r be a choen integer Suppoe that the matrix W C n m uing the rule from Propoition 11 Alo let the SVD factorization of W i of the general form V i choen W = UΣV (31) where U C n n and V C m m are column-orthogonal and Σ C n m i a diagonal matrix with the ingular value of W in decent order on the main diagonal Suppoe that the nonzero ingular value of W are ordered a σ 1 σ 2 σ (32) A it i known SVD decompoition (31) i not a full-rank factorization of W In order to generate the full-rank factorization of W ariing from (31) let u conider the matrice U C n n V C m m and Σ C n m partitioned in appropriate block U = [ [ [ Σ O U U R V = V V R Σ = O O where U C n V C m and Σ = diag {σ 1 σ } Then the compact SVD of W (33) W = U Σ V (34) i it full-rank factorization Suppoe that the QR factorization of W a in Theorem 3311 from [22 i of the form W = QRP (35) where P i an m m permutation matrix Q C n n Q Q = I n and R C n m Aume that P i choen o that Q and R can be partitioned a Q = [ Q Q R R = [ R O i an upper triangular matrix (36)

5 B I Shaini / Filomat 30:2 (2016) where Q (rep R ) conit of the firt column of the matrix Q (rep R) The column of Q form an orthonormal bai for R(W) and the column of Q R an orthonormal bai for N(W ) (ee [10 21) URV decompoition of the matrix W C n m r: [ C O W = URV = U V O O (37) where U = [ U U R C n n V = [ V V R C m m are orthogonal matrice and C C i invertible The matrice U and V i generated from the firt column of U and V repectively The column of U form an orthonormal bai for R(W) and the column of U R an orthonormal bai for N(W ) The column of V form an orthonormal bai for R(W ) and the column of V R an orthonormal bai for N(W) (ee [8 page ) Therefore uing known propertie of the matrix Q from the QR decompoition it i poible to conclude Q U Since a aid above the matrix U atifie U U = I we have [ [ U [ U U U R = U UU R U R U U R U = I R U R and o U U R = U R U = O U U = U R U R = I Similarly one can verify identitie V V R = V R V = O V V = V R V R = I In thi cae W = U (CV ) = Q CV (38) i a full rank factorization of W which i baed on it URV decompoition (37) 31 Computing {24} and {23}-invere by SVD Factorization Lemma 31 Aume that the matrix A Cr m n i given and 0 < r i a elected integer Aume that T i a ubpace of C n of dimenion r and S i a ubpace of C m of dimenion m Let G C m be an arbitrary matrix atifying N(G) = S N(GA) = T and rank(ga) = rank(g) = Let GA = U Σ V Σ = diag (σ 1 σ ) (39) be the compact SVD of GA Then A (24) TS = V diag ( σ 1 1 ) σ 1 U G (310) Proof Uing known SVD repreentation of the Moore-Penroe invere we obtain V diag ( σ 1 1 ) σ 1 U G = (GA) G According to Lemma 21 we get which i a verification of the identity (310) (GA) G = A (24) N(GA) N(G) = A(24) TS Lemma 32 Aume that the matrix A Cr m n i given and 0 < r i a elected integer Aume that T i a ubpace of C n of dimenion r and S i a ubpace of C m of dimenion m Let F C n be an arbitrary matrix atifying R(F) = T R(AF) = S and rank(af) = rank(f) = Let AF = U Σ V Σ = diag (σ 1 σ ) (311) be the compact SVD of AF Then A (23) TS = F V diag ( σ 1 1 ) σ 1 U (312)

6 B I Shaini / Filomat 30:2 (2016) In Theorem 31 we how that {2 4} and {2 3}-invere of precribed range and null pace can be computed without computation of the SVD of the matrix product GA or AF repectively Theorem 31 Let A Cr m n be the given matrix and r be a given integer (a) Let G be m matrix of rank atifying rank((ga) G) = and (34) be the full-rank factorization of W = (GA) G C n m implied by the compact SVD of W Then the following tatement hold: A (24) N(GA) N(G) = A(24) R(U )N(V ) = U (Σ V AU ) 1 Σ V = (WA) W = (GA) G A{2 4} (313) (b) Let F be n matrix of rank atifying rank(f(af) ) = If (34) i a full rank factorization of W = F(AF) C n m the following tatement are valid: A (23) R(F)N(AF) = A (23) R(U )N(V ) = U (Σ V AU ) 1 Σ V = W(AW) = F(AF) A{2 3} (314) Proof (a) In thi cae we have o that W = U (Σ V ) = (GA) G R(W) = R(U ) = R((GA) ) = N(GA) N(W) = N(Σ V ) = N(V ) = N(G) Later the identitie rank(σ V ) = rank(σ V A) = are atified Therefore condition from the part (3) of Lemma 33 from [3 are atified which implie A (2) N(GA) N(G) = U (Σ V AU ) 1 Σ V Since Σ V AU i a full-rank factorization of the invertible matrix Σ V AU the revere order low of the Moore-Penroe invere i applicable which implie Thi part of the proof can be completed uing A (2) N(GA) N(G) = (Σ V A) Σ V A{2 4} (Σ VA) Σ V = (U Σ VA) U Σ V = (WA) W = ((GA) GA) (GA) G = (GA) ( GA(GA) ) G = (GA) GA(GA) G = (GA) G (b) Thi part of the proof can be verified in a imilar way uing the part (2) of Lemma 33 from [3

7 B I Shaini / Filomat 30:2 (2016) A combined SVD of two matrice of identical dimenion ha been introduced with the motivation to avoid explicit computation of matrix product and quotient in the computation of the SVD of ome matrix product and quotient It i poible to ue the combined SVD to compute {2 3}-invere of A C m n r of the form F(AF) where F C n AF C m Intead of computation of the matrix product AF it eem more appropriate to ue a proper combination of truncated factorization of A of the order A = U A Σ A V = U A diag ( σ A 1 ) σa V (315) and compact SVD of F : F = U F Σ F V = U F diag ( σ F 1 σf ) V (316) In Theorem 32 we invetigate further propertie of the thin Moore-Penroe invere ( A ) which i introduced in [7 Theorem 32 Under the aumption of Theorem 31 the following tatement are valid: A = ( A ) = A(23) R(V )R(U F ) A {2 3} Proof Uing (315) and (316) we have ( ) ( ) ( ) (U A F = U A Σ A Σ F U F = U A diag σ G 1 σa 1 ) σg 1 σa F Thi further implie A = V diag 1 1 ( ) σ A σ U A A 1 ( ) (U = V diag σ F 1 ) σf F U F diag = F(A F) 1 σ F 1 σa 1 1 σ F σ A ( ) U A The proof can be completed uing R(F) = R(V ) N(F) = N (( U F ) ) = R ( U F ) 32 Computing {24} and {23}-invere by QR Factorization Propoition 31 [19 The following two tatement are valid: Q (R P AQ ) 1 R P = A (2) R(W)N(W) = A(2) R(Q )N(R P ) (317) Theorem 33 Let A C m n r Propoition 12 = Q (Q WAQ ) 1 Q W (318) be the given matrix r be a given integer and the matrice F G are choen a in (a) If (35) i a full rank factorization of W = (GA) G C n m the following i atified: Q (R P AQ ) 1 R P = (GA) G = A (24) N(GA) N(G) A{2 4} (319) (b) If (35) i a full rank factorization of W = F(AF) C n m the following hold: Q (R P AQ ) 1 R P = F(AF) = A (23) R(F)R(AF) A{2 3} (320)

8 B I Shaini / Filomat 30:2 (2016) Proof (a) In thi cae we have W = Q (R P ) = (GA) G (321) Therefore condition from the part (3) of Lemma 33 from [3 are atified which implie Thi part of the proof can be completed uing Q (R P AQ ) 1 R P = (R P A) R P A{2 4} (R P A) R P = (Q R P A) Q R P = ((GA) GA) (GA) G = (GA) G = A (2) N(GA) N(G) (b) Thi dual tatement can be verified in a imilar way uing the part (2) of Lemma 33 from [3 33 Computing {24} and {23}-invere by URV Factorization Theorem 34 Let A C m n r be the given matrix r be a given integer and the matrice F G are choen a in Propoition 12 (a) If (38) i a full rank factorization of W = (GA) G C n m the following i atified: A (24) N(GA) N(G) = Q (CV AQ ) 1 CV = Q ( Q WAQ ) 1 Q W = (GA) G A{2 4} (322) (b) If (38) i a full rank factorization of W = F(AF) C n m the following hold: A (23) N(GA) N(G) = Q (CV AQ ) 1 CV = F(AF) A{2 3} (323) Proof (a) In thi cae we have W = U (CV ) = Q CV = (GA) G Therefore condition from the part (3) of Lemma 33 from [3 are atified which implie From (38) we obtain and later Q (CV AQ ) 1 CV = (U CV A) U CV A{2 4} CV = Q W Q (CV AQ ) 1 CV = Q ( Q WAQ ) 1 Q W which confirm the econd identity in (322) The econd identity in (322) can be verified uing Q (CV AQ ) 1 CV = (U CV A) U CV = ((GA) GA) (GA) G = (GA) G = A (2) N(GA) N(G) (b) Thi part of the proof can be verified in a imilar way uing the part (2) of Lemma 33 from [3

9 B I Shaini / Filomat 30:2 (2016) Algorithm In the equel we tate the next unique algorithm for generating A (24) and A(23) invere of a given matrix TS TS A by mean of three matrix decompoition Algorithm 41 Computing the A (24) TS invere of the matrix A uing the complete orthogonal factorization of W = (GA) G (Algorithm COFATS24) or W = F(AF) (Algorithm COFATS23) Require: The matrix A of dimenion m n and of rank r Require: The matrix G of dimenion m and of rank or the matrix F of dimenion n and of rank 1: Compute the matrix W = (GA) G or the matrix W = F(AF) 2: Compute the SVD QR or URV decompoition of the matrix W in the form (31) (35) or (37) repectively 3: Generate the full rank decompoition of the matrix W a in (34) (321) or (38) repectively 4: Solve one of the following matrix equation: - In the cae of SVD decompoition olve Σ V AU X = Σ V - In the cae of QR decompoition olve R P AQ X = R P or Q WAQ X = Q W - In the cae of URV decompoition olve CV AQ X = CV 5: Compute the output: - In the cae of SVD decompoition return the matrix U X - In the cae of QR or URV decompoition return the matrix Q X We preent a comparion of Algorithm 41 with the modified Succeive Matrix Squaring (SMS) iterative algorithm for computing {2 3} and {2 4}-invere from [16 The main difference between the modified SMS algorithm and Algorithm 41 are contained in the following: 1 The modified SMS algorithm i iterative while Algorithm 41 i a direct method for computing generalized invere 2 Moreover QR variant of Algorithm 41 i applicable in exact computation of generalized invere of matrice whoe entrie are rational number More preciely the programming package Mathematica yield the QR decompoition for a numerical matrix in the form of a matrix with rational entrie Later uing the exact computation poibilitie in Mathematica it i poible to produce the exact output of Algorithm 41 About the package Mathematica ee for example [23 Let u mention that it i poible to force Mathematica to give an approximate numerical output by ending the uer input with //N 3 Roundoff error accumulate during the iterative tep of the SMS method Sometime thee roundoff error can endanger it convergence In thee cae direct method are appropriate choice 4 Finally in our numerical experiment we oberved an additional drawback of the modified SMS algorithm Namely the optimal value β C opt of the parameter β (from [16) i very mall which caue a very large number of iterative tep (approximately ) Thi choice caue the collape of the SMS algorithm 5 Numerical Example Numerical example are performed in the programming package Mathematica In both Example 51 part (b) and Example 52 part (b) we derive exact generalized invere

10 Example 51 The tarting matrix in thi example i and the matrix G i equal to B I Shaini / Filomat 30:2 (2016) G = A = [ a) The SVD decompoition of the matrix W = (GA) G aociated with the = 2 larget ingular value of W i defined by W = U 2 Σ 2 V 2 where {U 2 Σ 2 V 2 } = [ The repreentation (313) give the following {2 4} invere of A: X 1 = A (23) N(GA) N(G) = U 2(Σ 2 V 2 AU 2) 1 Σ 2 V = = (GA) G b) The QR decompoition of W = (GA) G i determined by {Q R P} = I The repreentation (317) produce the following {2 4} invere of A: A (24) N(GA) N(G) = Q (R P AQ ) 1 R P = Real approximation of the rational matrix A (24) N(GA) N(G) i identical to X 1

11 Alo imple verification how that B I Shaini / Filomat 30:2 (2016) (WA) W = (GA) G = X 1 c) The modification of the SMS method from [16 produce the following approximate outer invere after 40 iteration: X SMS = But after 50 iterative tep we get the wrong reult: After 60 iterative tep we oberved beginning of the divergence: Let u oberve that the approximation of the exact outer invere with 10 decimal i equal to It i eay to oberve that the bet poible reult produced by the modified SMS method denoted by X SMS i a much wore approximation than the approximation X 1 produced by the SVD factorization To verify thi tatement we compute correponding matrix norm: A (24) N(GA) N(G) X 1 = A (24) N(GA) N(G) X SMS = Example 52 Let u chooe the ame matrix A a in Example 51 and the matrix F equal to F = The matrix W appropriate to A i now given by W = F(AF) a) The SVD decompoition of W = F(AF) i {U 2 Σ 2 V 2 } = [

12 The repreentation (314) give the following {2 3} invere of A: B I Shaini / Filomat 30:2 (2016) X 2 = A (23) N(GA) N(G) = U (Σ VAU ) 1 Σ V = Let u mention that F(AF) i the rational matrix whoe approximation i jut the matrix X b) The QR decompoition of W = F(AF) i given by {Q 2 R 2 P} = I 6 The repreentation (317) give the following exact {2 3} invere of A: A (23) N(GA) N(G) = Q 2(R 2 P AQ 2 ) 1 R 2 P = Again imple verification how that X 2 i a real approximation of Q 2 (R 2 P AQ 2 ) 1 R 2 P Example 53 In thi example we conider outer invere of the invertible 8 8 matrix generated applying a = 1 and n = 8 on the tet matrix S n from [27 A =

13 B I Shaini / Filomat 30:2 (2016) initiated by the randomly generated 2 8 matrix G = [ The matrix W = (GA) G hould be ued in contruction of correponding {2 4}-invere of A a) SVD decompoition W = U 2 Σ 2 V of W i defined by 2 {U 2 Σ 2 V 2 } = [ The repreentation (317) produce the following {2 4} invere of the invertible matrix A: X 3 = b) On the other hand QR decompoition of W i defined by the triple of matrice {Q R P} = I 8

14 B I Shaini / Filomat 30:2 (2016) The matrix R 2 P AQ 2 equal to i almot ingular and it inverion produce computational error which caue that the generated approximation (319) of outer invere doe not atify the matrix equation (2) Example 54 In thi example we verify the reult of Theorem 32 Conider the matrix A = defined by A = U A 2 ΣA 2 V 2 where [ ( ) ( and the matrix F defined by F = V 2 Σ F 2 U F 2) where After a verification we get (A ) = [ = F(AF) Example 55 In thi example we compare Algorithm 41 and the modified SMS algorithm with 10 iterative tep Matrice A of the order m n where m = 30 k n = 10 k and of rank r = n/2 Matrice A of the order n m and of rank r = n/3 By X SVD X QR X SMS we denote outer invere produced by the SVD decompoition QR decompoition and the modified SMS iterative proce The trike in Table 1 mean that the modified SMS method produce the incorrect zero matrix in the output X SMS = 0

15 B I Shaini / Filomat 30:2 (2016) Dimenion X SVD AX SVD X SVD X QR AX QR X QR X SMS AX SMS X SMS Table 1: CPU time on the tet block matrix V n 6 Concluion In the current paper we further invetigate condition for the exitence repreentation and algorithm for computing {2 4} and {2 3}-invere with precribed range and null pace We alo derive correponding repreentation ariing from SVD QR and URV matrix decompoition An unique algorithm for computing {2 4} and {2 3}-invere i defined Our numerical experience how that direct method defined in 41 produce reult with ignificantly better preciion than the iterative method propoed in [16 Reference [1 A Ben-Irael TNE Greville Generalized invere: theory and application Second Ed Springer 2003 [2 C-Guang Cao X Zhang The generalized invere A (2) and it application J Appl Math Comput 11 (2003) T [3 YL Chen X Chen Repreentation and approximation of the outer invere A (2) of a matrix A Linear Algebra Appl 308 (2000) TS [4 RE Cline Invere of rank invariant power of a matrix SIAM J Numer Anal 5 (1968) [5 DS Cvetković-Ilić PS Stanimirović M Miladinović Comment on ome recent reult concerning {2 3} and {2 4}-generalized invere Appl Math Comput 218 (2011) [6 DS Djordjević PS Stanimirović General repreentation of peudoinvere Mat Venik 51 (1999) [7 BI Shaini F Hoxha Computing generalized invere uing matrix factorization Facta Univeritati (Niš) Ser Math Inform 28 (2013) [8 C Meyer Matrix Analyi and Applied Linear Algebra SIAM Philadelphia 2000 [9 B de Moor Generalization of the ingular value and QR decompoition Signal Proceing 25 (1991) [10 R Piziak P L Odell Full Rank Factorization of Matrice Mathematic Magazine Vol 72 No 3 (Jun 1999) pp [11 CR Rao SK Mitra Generalized Invere of Matrice and it Application John Wiley and Son Inc New York London Sydney Toronto 1971 [12 X Sheng G Chen Full-rank repreentation of generalized invere A (2) and it application Comput Math Appl 54 (2007) TS [13 PS Stanimirović Application of hyper-power method for computing matrix product Univ Beograd Publ Elektrotehn Fak Ser Mat 15 (2004) [14 PS Stanimirović Block repreentation of {2} {12} invere and the Drazin invere Indian J Pure Appl Math 29 (1998) [15 PS Stanimirović DS Cvetković-Ilić Succeive matrix quaring algorithm for computing outer invere Appl Math Comput 203 (2008) [16 PS Stanimirović D Cvetković-Ilić S Miljković M Miladinović Full-rank repreentation of {24} {23}-invere and ucceive matrix quaring algorithm Appl Math Comput 217 (2011) [17 PS Stanimirović DS Djordjević Full-rank and determinantal repreentation of the Drazin invere Linear Algebra Appl 311 (2000) [18 PS Stanimirović D Pappa VN Katiki VN IP Stanimirović Computation of A (2) -invere uing QDR factorization Linear TS Algebra Appl 437 (2012) [19 PS Stanimirović D Pappa VN Katiki IP Stanimirović Full-rank repreentation of outer invere baed on the QR decompoition Appl Math Comput 218 (2012) [20 PS Stanimirović M Taić Computing generalized invere uing LU factorization of matrix product Int J Comput Math 85 (2008) [21 G Wang Y Wei S Qiao Generalized invere: theory and computation Science Pre 2003 [22 D Watkin Fundamental of Matrix Computation Wiley-Intercience New York 2002

16 B I Shaini / Filomat 30:2 (2016) [23 S Wolfram The Mathematica Book Fifth Edition Wolfram Media 2003 [24 H Yang D Liu The repreentation of the generalized invere A (23) and it application J Comput Appl Math 224 (2009) TS [25 H Yang D Liu J Xu Matrix left ymmetry factor and it application in generalized invere A (24) Appl Math Comput 197 (2008) TS [26 B Zheng L Ye DS Cvetković-Ilić Generalized invere of a normal matrix Appl Math Comput 206 (2008) [27 G Zielke Report on tet matrice for generalized invere Computing 36 (1986)