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2 pplied Matematics and Computation 90 (2007) bstract Generalized LM-inverse of a matrix augmented by a column vector Firdaus E. Udwadia a, *, Pailaung Poomsiri b a erospace and Mecanical Engineering, Civil Engineering, Systems rcitecture Engineering, and Information and Operation Management, 430 K Olin Hall of Engineering, University of Soutern California, Los ngeles, C , US b Postdoctoral Researc ssociate, erospace and Mecanical Engineering, University of Soutern California, Los ngeles, C , US Tis paper presents a direct proof of te recursive formulae for te generalized LM- inverse of a matrix augmented by a column vector. Te recursive relations are proved by direct verification of te four conditions of te generalized LM-inverse. Several auxiliary results pertaining to generalized inverses are also provided. Ó 2006 Elsevier Inc. ll rigts reserved. Keywords: Generalized inverse; Moore Penrose M-Inverse; Moore Penrose LM-Inverse; Recursive formulae; Least squares problem. Introduction Let us begin by considering a set of linear equations Bx b; were B is an m by n matrix, b is an m-vector, and x is an n-vector. Te generalized LM-inverse of te matrix B is te matrix suc tat te solution x B þ LM b minimizes bot G L =2 ðbx bþ 2 kbx bk 2 L ð3þ and H M =2 x 2 kk x 2 M ; were L is an m by m symmetric positive definite matrix and M is an n by n symmetric positive definite matrix. ðþ ð2þ ð4þ * Corresponding autor. address: fudwadia@usc.edu (F.E. Udwadia) /$ - see front matter Ó 2006 Elsevier Inc. ll rigts reserved. doi:0.06/j.amc

3 000 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) Below are te four conditions for te generalized LM-inverse []. ðiþ BB þ LM B B; ðiiþ B þ LM BBþ LM Bþ LM ; ðiiiþ LBB þ LM is symmetric; ð7þ ðivþ MB þ LMB is symmetric: ð8þ We note tat te generalized LM-inverse is te more general kind of te Moore Penrose inverse. Te concept of Moore Penrose (MP) inverses was first introduced by Moore [2] in 920 and later independently by Penrose [3] in 955. In 960, Greville [4] gave te first formulae for recursively determining te Moore Penrose inverse of a matrix. His algoritm provides an update of te MP inverse of a matrix wenever new information becomes available. s a result, te recursive formulae ave found extensive use in many areas of applications. mong tem are statistical inference [5], filtering teory, estimation teory [6], system identification [7], optimization and control, and most recently analytical dynamics [8,9]. In 997 Udwadia and Kalaba [0] provided an alternative and simple constructive proof of Greville s formulae, and later [,2] developed recursive relations for different types of generalized inverses of a matrix including te least-squares generalized inverse, te minimum-norm generalized inverse, and te Moore Penrose (MP) inverse of a matrix. Recently, te recursive formulae for te generalized M-inverse [3,4] and for te generalized LM-inverse were obtained. Tose for te generalized LM-inverse were proved constructively [5]. In tis paper, we provide a muc simpler and alternative proof for te recursive formulae of te generalized LM-inverse, B þ LM,of any given matrix, B, partitioned as B ½ j a Š, were is an m by n matrix and a is a column vector of m components. We sow tat te four conditions of te generalized LM-inverse of te recursive formulae are satisfied. Besides its inerent simplicity, our proof requires several subsidiary properties of te generalized LM-inverse of a matrix, many of wic appear to be ereto unknown; tey are presented in te ppendix. More general tan te generalized M-inverse, te generalized LM-inverse finds use in an even wider range of application areas tan te Moore Penrose inverse areas ranging from system teory, statistics, filtering, control teory, and optimization, to signal processing and mecanics. 2. Recursive formulae of te generalized LM-inverse of a matrix augmented by a column vector 2.. Result For any given matrix B ½ j a Š ð9þ its generalized LM-inverse formulae are given by B þ LM ½ j a Šþ LM þ þ ad þ L ; wen d ði þ d þ Þa 6 0; ð0þ L þ þ a p ; wen d ði þ LM Þa 0; ðþ were is an m by (n ) matrix, a is a column vector of m components, d þ L dt L=ðd T LdÞ, b qt MU, b = q T Mq, U þ, q v þ p, p ði þ 0 m ÞM ~m; and v þ M a. Note tat L is a symmetric positive definite m by m matrix, and M M ~m ~m T ; ð2þ m were M is a symmetric positive definite n by n matrix, M is te symmetric positive definite (n ) by (n ), ~m is te column vector of (n ) components, and m is te scalar. ð5þ ð6þ

4 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) It sould be noted tat te formulae are given in two separate cases; wen d 5 0 and wen d = 0. Wen d 5 0, te added column vector a is not a linear combination of te columns of, and wen d = 0, te added column vector a is a linear combination of te columns of (see ppendix in Ref. [5] for a proof). Proof. Case : (wen d 5 0) Because te BB þ M and Bþ MB are repetitively used to verify all four properties of te generalized LM-inverse, we sall first evaluate BB þ M and Bþ MB. By Eqs. (9) and (0), we ave BB þ LM ½ j a Š þ þ ad þ L þ d þ þ ad þ L ðpþdþ L þ adþ L : ð3þ L Since p = 0 (see Property in ppendix) and d ði þ Þa, we get BB þ LM þ þ ad þ L þ adþ L þ þði þ Þad þ L þ þ dd þ L : ð4þ gain by Eqs. (9) and (0), we obtain B þ LM B þ þ ad þ L ½ j a Š ð5þ d þ L þ þ aðd þ L Þpðdþ L Þ þ a þ aðd þ L aþpðdþ L aþ d þ L d þ L a : ð6þ Using te relations d þ L 0 and dþ L a (see Properties 4 and 5 in ppendix), we ave B þ LM B þ þ a þ a p 0 þ p 0 : ð7þ We now verify te four properties of te generalized LM-inverse. Generalized LM-inverse condition : BB þ LM B B Using Eqs. (9) and (4), we obtain BB þ LM B ðbbþ LM ÞB ðþ þ dd þ L Þ½ j a Š ½þ þ dðd þ L Þ j þ a þ dðd þ L aþ Š: ð8þ Because þ, d þ L 0, dþ L a (see Properties 4 and 5 in ppendix), and d ði þ Þa, we ave BB þ LM B þ a þði þ Þa ½ j a Š B: Generalized LM-inverse condition 2: B þ LM BBþ LM Bþ LM Using Eqs. () and (4), we get B þ LM BBþ LM Bþ LM ðbbþ LM Þ þ þ ad þ L ½ þ d þ þ dd þ L Š; ð9þ L " þ þ þð þ dþd þ L þ aðd þ L Þþ þ ad þ L ddþ L # pðdþ L Þþ pd þ L ddþ L ðd þ L Þþ þ d þ : ð20þ L ddþ L Since þ þ þ, þ d 0, d þ L 0 (see Properties 3 and 4 in ppendix), and dþ L ddþ L dþ L,we obtain B þ LM BBþ LM þ þ ad þ L B þ LM : d þ L Generalized LM-inverse condition 3: LðBB þ LMÞ is symmetric Since L þ and Ldd þ L are symmetric, using Eq. (4), we ave LðBB þ LM ÞLðþ þ dd þ L ÞLþ þ Ldd þ L ; wic is symmetric.

5 002 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) Generalized LM-inverse condition 4: MðB þ LMBÞ is symmetric Using Eqs. (2) and (7), we obtain MðB þ LM BÞ M ~m þ ~m T m 0 " p M þ # M p þ ~m ~m T þ ~m T p þ m E ; E ;2 ; ð2þ E 2; E 2;2 were E,, E,2, E 2,, ande 2,2 represent te elements (,), (, 2), (2,), and (2, 2) of te matrix MðB þ LM BÞ, respectively. Note tat E, is te (n ) by (n ) matrix, E,2 is te column vector of (n ) components, E 2, is te row vector of (n ) components, and E 2,2 is te scalar. We see tat E ; M ð þ Þ is symmetric since þ is te generalized -inverse of, wile E 2,2 is a scalar, and is terefore symmetric. Tus, for MðB þ LM BÞ to be symmetric, we need to sow tat E,2 is te transpose of E 2,. Using p ði þ ÞM ~m, te element E,2 can be written as M p þ ~m M ði þ ÞM ~m þ ~m M ð þ ÞM ~m ðþ Þ T ~m; ð22þ wic is te transpose of te element E 2,. Hence, MðB þ LMBÞ is symmetric. We ave sown tat all four generalized LM-inverse conditions are satisfied. Hence, te formula (0) is verified. Case 2: (wen d =0) We begin again by evaluating BB þ LM and Bþ LMB quantities tat we will need furter along. Using Eqs. (9) and (), weave BB þ LM ½ j a Š þ þ a p ½ þ LM ð þ aþ ðpþ þ aš: ð23þ Since þ a a and p = 0 (see Properties and 2 in ppendix), we get BB þ LM ½þ a þ aš þ : Using Eqs. (9) and (), weave B þ LM B þ þ a p ½ j a Š þ þ a p þ a þ aa pa : a ð25þ We next verify te four properties of te Generalized LM-inverse. Generalized LM-inverse condition : BB þ LM B B Using te fact tat þ and þ a a (see Property 2 in ppendix), by Eqs. (9) and (24) we obtain BB þ LM B ðbbþ LM ÞB þ ½ j a Š ½ þ þ a Š½ j a Š B: Generalized LM-inverse condition 2: B þ LM BBþ LM Bþ LM By Eqs. () and (24), we ave B þ LM BBþ LM Bþ LM ðbbþ LM Þ þ þ " # a p þ þ LM LM þ þ að þ Þpð þ Þ þ : ð26þ Since þ þ þ and þ (see Property 9 in ppendix), we ave B þ LM BBþ LM þ þ a p B þ LM : ð24þ

6 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) Generalized LM-inverse condition 3: LðBB þ LMÞ is symmetric Because L þ is symmetric, by Eq. (24) we ave ðbb þ LM ÞT ð þ Þ T L þ L LBB þ LM L : Generalized LM-inverse condition 4: MðB þ LMBÞ is symmetric. Using Eqs. (2), (25), andv þ a, we obtain MðB þ LM BÞ M ~m þ v p v va pa ~m T m ; a " M ð þ # v pþþ~mðþ M ðv va paþþ~mðaþ ~m T ð þ v pþþmðþ ~m T ðv va paþþmðaþ E ; E ;2 ; ð27þ E 2; E 2;2 were E,, E,2, E 2,,andE 2,2 represent te elements (,), (, 2), (2,), and (2,2) of te matrix MðB þ LM BÞ, respectively. Note tat E, is te (n ) by (n ) square matrix, E,2 is te column vector of (n ) components, E 2, is te row vector of (n ) components, and E 2,2 is te scalar. For MðB þ LMBÞ to be symmetric, we need to sow tat E, is symmetric and E,2 is te transpose of E 2,. Let us first sow tat E, is symmetric. We can rewrite E, as E ; M þ ðm v þ M p ~mþðþ: Since M v þ M p ~m bðþ T (see Property 2 in ppendix), we get E ; M þ bðþ T ðþ: Because M þ and b() T () are symmetric, E, is symmetric. Next, we will sow tat E,2 is te transpose of E 2,. Let us rewrite E,2 as E ;2 M v ðm v þ M p ~mþðaþ: Using a b ðvt M v ~m T vþ and M p ~m ð þ Þ T ~m (see Properties 0 and in ppendix) in Eq. (30), we obtain i E ;2 M v M v ð þ Þ T ~m v T M v ~m T v b : ð3þ On te oter and, E 2, can be written as E 2; ~m T þ ~m T v þ ~m T p m ðþ: ð32þ Using b ðvt M ~m T þ Þ and ~m T v þ ~m T p m v T M v v T ~m b (see Properties 8 and 4 in ppendix) in Eq. (32), we get E 2; ~m T ð þ Þ v T M v v T ~m b v T M ~m T þ b ; ð33þ wic can be simplified to E 2; v T M v T M v v T ~m v T M ~m T þ b : ð28þ ð29þ ð30þ ð34þ It can be seen from Eqs. (3) and (34) tat E,2 is te transpose of E 2,. Since we ave already sown tat E, is symmetric and since E 2,2 is scalar wic is symmetric, te symmetry of MðB þ LMBÞ is verified. We ave sown tat all four generalized LM-inverse conditions are satisfied. Hence, te formula () is verified.

7 004 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) Conclusions Te recursive formulae for obtaining te generalized LM-inverse of any general matrix augmented by a column vector were first given in Ref. 5. Tere tey were derived in a constructive manner. We erein provide an alternative proof of teir formulae by directly verifying tat te four conditions of te generalized LM-inverse are satisfied, tereby confirming te validity of te formulae, and providing several new auxiliary results related to tese generalized inverses. ppendix Tis section provides some properties tat are used for verifying te recursive formulae for determining of te generalized LM-inverse of a matrix. Property. p = 0. Proof. Since p ði þ ÞM ~m, we ave p ði þ ÞM ~m 0: Property 2. a þ a (wen d = 0). Proof. Because d ði þ Þa 0, we get a þ a: Property 3. þ d 0. Proof. Since d ði þ Þa, we obtain þ d þ ði þ Þa 0: Property 4. d þ L 0. Proof. Since d ði þ Þa and d þ L ðdt LdÞ d T L, we ave d þ L ðdt LdÞ d T L ðd T LdÞ ði þ T Þa L ðd T LdÞ a T ði þ Þ T L ðd T LdÞ a T LðI þ ÞL L 0: Property 5. d þ L a. Proof. Using d þ L ðdt LdÞ d T L and d ði þ Þa, weave d þ L a dt La d T Ld ½ðI þ ÞaŠ T La ½ðI þ ÞaŠ T L½ðI þ ÞaŠ a T þ LðI ÞL La a T LðI þ ÞL LðI þ Þa at LðI þ Þa a T LðI þ Þa : Property 6. b ðvt M ~m T Þ þ. Proof. Since b qt MU, were b = q T Mq, q and U,weave þ 0 m v þ p, v þ a, p ði þ ÞM ~m, M M ~m ~m T m,

8 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) b qt MU T v þ p M ~m þ b ~m T m 0 m b vt M þ þ p T M þ ~m T þ : b vt þ p T j " # M þ ~m T þ ; Because p T M þ ½~m T M ði þ Þ T ŠM þ ~m T M M ði þ ÞM M þ 0, we obtain b ðvt M þ ~m T þ Þ b ðvt M ~m T Þ þ : Property 7. v T M þ v T M. Proof. Since v þ a and M þ ð þ Þ T M, we ave v T M þ ð þ aþ T ð þ Þ T M ð þ þ aþ T M ð þ aþ T M v T M : Property 8. b ðvt M ~m T þ Þ. Proof. Since b ðvt M ~m T Þ þ b ðvt M þ ~m T þ Þ b ðvt M ~m T þ Þ: Property 9. þ. and v T M þ v T M (see Properties 6 and 7 above), we ave Proof. Since b ðvt M ~m T Þ þ (see Property 6 above) and þ þ þ, we get þ b ðvt M ~m T Þ þ þ b ðvt M ~m T Þ þ : Property 0. a b ðvt M v ~m T vþ. Proof. Because þ a v and b ðvt M ~m T Þ þ a b ðvt M þ a ~m T þ aþ b ðvt M v ~m T vþ: Property. M p ~m ð þ Þ T ~m. Proof. Since p ði þ ÞM ~m, we get (see Property 6 above), we ave M p ~m M ½ðI þ ÞM ~mš~m M þ M ~m ðþ Þ T ~m: Property 2. M v þ M p ~m ½M v ð þ Þ T ~mš T bðþ T. Proof. Since M p ~m ð þ Þ T ~m and v T M ~m T þ b (see Properties 8 and above), we get M v þ M p ~m M v ð þ Þ T ~m ½v T M ~m T þ Š T bðþ T : Property 3. p T M ~m T ði þ Þ. Proof. Because p ði þ ÞM ~m and v þ a,weave p T M ½ðI þ ÞM ~mšt M ~m T M ½M ði þ ÞM ŠM ~m T ði þ Þ:

9 006 F.E. Udwadia, P. Poomsiri / pplied Matematics and Computation 90 (2007) Property 4. ~m T v þ ~m T p m v T M v v T ~m b. Proof. Since q v þ p, and M M ~m ~m T, we ave m b q T Mq v þ p T M ~m v þ p ~m T v T M v þ 2p T M v 2~m T v þ p T M p 2~m T p þ m; m were we ave used p T Mv = v T Mp, ~m T v v T ~m, and ~m T p p T ~m since tey are scalars. Using p T M ~m T ði þ Þ (see Property 3 above), we ave p T M v ~m T ði þ Þ þ a 0 and p T M p ½~m T ði þ ÞŠ½ðI þ ÞM ~mš ~mt ½ðI þ ÞM ~mš ~mt p. Tus, we obtain b v T M v 2~m T v ~m T p þ m, wic gives m b v T M v þ 2~m T v þ ~m T p: Using te relation above and again te fact tat ~m T v v T ~m, we ave ~m T v þ ~m T p m ~m T v þ ~m T p ðbv T M v þ 2~m T v þ ~m T pþv T M v v T ~m b: References [] C.R. Rao, S.K. Mitra, Generalized Inverses of Matrices and Its pplications, Wiley, New York, 97. [2] E.H. Moore, On te reciprocal of te general algebraic matrix, Bulletin of te merican Matematical Society 26 (920) [3] R.. Penrose, Generalized inverse for matrices, Proceedings of te Cambridge Pilosopical Society 5 (955) [4] T.N.E. Greville, Some applications of te pseudoinverse of a matrix, SIM Review 2 (960) [5] C.R. Rao, note on a generalized inverse of a matrix wit applications to problems in matematical statistics, Journal of te Royal Statistical Society Series B 24 (962) [6] F. Graybill, Matrices and pplications to Statistics, second ed., Wadswort, Belmont, California, 983. [7] R.E. Kalaba, F.E. Udwadia, ssociative memory approac to te identification of structural and mecanical systems, Journal of Optimization Teory and pplications 76 (993) [8] F.E. Udwadia, R.E. Kalaba, nalytical Dynamics: New pproac, Cambridge University Press, Cambridge, England, 996. [9] J. Franklin, Least-squares solution of equations of motion under inconsistent constraints, Linear lgebra and Its pplications 222 (995) 9 3. [0] F.E. Udwadia, R.E. Kalaba, n alternative proof of te Greville formula, Journal of Optimization Teory and pplications 94 (997) [] F.E. Udwadia, R.E. Kalaba, unified approac for te recursive determination of generalized inverses, Computers and Matematics wit pplications 37 (999) [2] F.E. Udwadia, R.E. Kalaba, Sequential determination of te {,4}-inverse of a matrix, Journal of Optimization Teory and pplications 7 (2003) 7. [3] F.E. Udwadia, P. Poomsiri, Te recursive determination of te generalized Moore Penrose M-inverse of a matrix, Journal of Optimization Teory and pplications 27 (3) (2005) [4] P. Poomsiri, B. Han, n alternative proof for te recursive formulae for computing te Moore Penrose M-inverse of a matrix, pplied Matematics and Computation 74 (2006) [5] F.E. Udwadia, P. Poomsiri, Recursive formulae for te generalized LM-inverse of a matrix, Journal of Optimization Teory and pplications 3 () (2006) 6.