SYMMETRIC POSITIVE SEMI-DEFINITE SOLUTIONS OF AX = B AND XC = D

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1 Joural of Pure ad Alied Mathematics: Advaces ad Alicatios olume, Number, 009, Pages SYMMERIC POSIIE SEMI-DEFINIE SOLUIONS OF AX B AND XC D School of Mathematics ad Physics Jiagsu Uiversity of Sciece ad echology Zhejiag, 003 P. R. Chia yuayx_703@63.com Abstract I this aer, a sufficiet ad ecessary coditio for the matrix equatios AX B ad XC D, where A R, B R, C R, ad D R, to have a commo symmetric ositive semi-defiite solutio X is established, ad if it exists, a reresetatio of the solutio set S X is give. A otimal aroximatio betwee a give matrix X R ad the affie subsace S X is discussed, a exlicit formula for the uique otimal aroximatio solutio is reseted, ad a umerical examle is rovided. 000 Mathematics Subject Classificatio: 5A09, 5A4. Keywords ad hrases: matrix equatio, symmetric ositive semi-defiite solutio, sigular value decomositio, otimal aroximatio. Received October 0, 009. Itroductio I this aer, we shall adot the followig otatio. the set of all m real matrices, symmetric ositive semi-defiite matrices i R deotes SPR deotes the set of all. I R reresets the idetity matrix of size., A, ad A stad for the trasose, the A 009 Scietific Advaces Publishers

2 00 Moore-Perose geeralized iverse, ad the Frobeius orm of a real matrix A, resectively. For A, B R, we defie a ier roduct i R : A, B trace( B A ), the orm R is a Hilbert sace. he matrix iduced by the ier roduct is the Frobeius orm. We write A 0, if A is a real symmetric ositive semi-defiite matrix. Matrix equatio is oe of the imortat study fields of liear algebra. he liear matrix equatios AX B, XC D, () have bee cosidered by may authors. I [6], Mitra gave the commo solutio of miimum ossible rak based o the geeralized matrix iverses ad the matrix rak. Li [4] discussed the geeralized reflexive solutio of (), a ecessary ad sufficiet coditio for the solvability ad the exressio of the geeral solutio were obtaied. Qiu [7] cosidered the costrait PX ± XP solutio of (), where P is a give Hermitia matrix satisfyig P. Qiu [8] further studied the least-squares I solutios to the Equatios () with some costraits: orthogoality, symmetric orthogoality, symmetric idemotet. Li [5] foud a sufficiet ad ecessary coditio for the matrix Equatios () to have symmetric ad skew-atisymmetric solutios over the real quaterio ad, for the cosistet case, rovided a reresetatio of its geeral solutio. Wag [9] cosidered the bisymmetric solutios of () over the real quaterio algebra. Recetly, Dajić [] studied the ositive solutios to the Equatios () for Hilbert sace oerators usig geeralized iverses, ad a sufficiet ad ecessary coditio for its solvability, ad a reresetatio of its geeral solutio was also established therei. I the reset aer, we will cosider symmetric ositive semi-defiite solutios of the matrix Equatios (), where A R, B R, C R, ad D R, ad a associated otimal aroximatio roblem: mi X S X X X, ()

3 SYMMERIC POSIIE SEMI-DEFINIE 0 where X is a give matrix i R ad S X is the solutio set of the matrix Equatios (). Clearly, whe X 0, the solutio of () is the miimum orm solutio of (). Usig the sigular value decomositio, we give a ecessary ad sufficiet coditio for the Equatios () to have a solutio X SPR, ad costruct the solutio set S X exlicitly, whe it is oemty. We show that there exists a uique solutio to the matrix otimal aroximatio roblem (), if the set S X is oemty ad reset a exlicit formula for the uique solutio.. he Solutio of the Matrix Equatios () o begi with, we itroduce a lemma [0]. Lemma. Let A R, B R, ad the sigular value decomositio of A be A 0 U, 0 0 (3) where U [ U, U ], [, ] are all orthogoal matrices ad the artitios are comatible with the size of diag{ σ, L, σ } > 0 t rak ( A). he, the matrix equatio has a solutio X SPR, if ad oly if t, AX B, (4) BA 0, rak( B) rak( BA ). (5) I which case, the geeral solutio of the equatio of (4) ca be exressed as where X X0 G, (6)

4 0 X 0 A A B ( I A A )( A B ) ( I A A ) B ( AB ) B( I A ), ad G is a arbitrary symmetric ositive semi-defiite matrix. Isertig (6) ito XC D, we get G C 0C (7) D X. (8) It is easily see that the equatio of (8) is equivalet to G ( D X C ) 0, (9) 0 ( D X ). (0) C 0C Sice, to A A ad AX 0 B, the the relatio of (9) is equivalet Let the sigular value decomositio of AD BC. () C be Ω 0 C P Q, 0 0 () where P [ P, P ], Q [ Q, Q ] are all orthogoal matrices ad the artitios are comatible with the size of Ω diag{ ω, L, ω } > 0, s rak( C ). It follows from Lemma that the equatio of (0) has a solutio ( t) ( t G SPR ), if ad oly if C C Notice that ( D X C ), rak(( D X C ) ) rak(( D X C ) ). ( D X C ) ( I A A )( D X C ) D X C herefore, the relatios of (3) ca be simlified as C C s (3) ( D X C ), rak(( D X C ) ) rak(( D X C ) ). (4)

5 SYMMERIC POSIIE SEMI-DEFINIE 03 Whe the coditios (4) hold, the geeral solutio of (0) ca be writte as where G G P WP 0, G0 DC ( DC ) ( I CC ) ( I CC ) D( D C ) D ( I CC ), t t t C C, D ( D X ), ad W 0 is a arbitrary matrix. 0C (5) As a summary of the above discussio, we have roved the followig result. heorem. Give A R, B R, C R, ad D R. Let the sigular value decomositios of A ad C be give by (3) ad (), resectively. he, the matrix Equatios () have a solutio X SPR, if ad oly if the coditios (5), (), ad (4) are satisfied. I this case, the solutio set of () ca be exressed as S { X SPR X X P WP }, (6) X where X X0 G0, ad W is a arbitrary symmetric ositive semi-defiite matrix. 3. he Solutio of the Otimal Aroximatio Problem () I order to solve the otimal aroximatio roblem (), we eed the followig lemma [3]. Lemma. Let E R, E ( E E ), ad let the olar decomositio of E be E FH, where F is a orthogoal matrix ad H is a symmetric ositive semi-defiite matrix. Let E ˆ ( E ), H the Eˆ E mi K SPR K E.

6 04 heorem. If the solutio set S X is oemty, the the otimal aroximatio roblem () has a uique solutio Xˆ S X. Furthermore, let J P ( X X ) P P XP ad the olar decomositio of J be J LM, (7) where L is a orthogoal matrix ad M 0. Let W ˆ ( J M ), the the uique solutio Xˆ of () ca be exressed as X ˆ X ˆ P WP. (8) Proof. By heorem, we kow that if the coditios (5), (), ad (4) are satisfied, the the solutio set S X is oemty. It is easy to verify that S X is a closed covex set i Hilbert sace SPR. herefore, for a give matrix ˆ X R, it follows from the best aroximatio theorem (see Aubi []), that there exists a uique solutio Xˆ i S such that Xˆ X mi X X. For ay matrix X S X, we have X X S X X X PWP ( X X ) P WP ( X X ) ξ PWP ( X X ) ξ P P WP P P ( X X ) P π W P P ( X X ), where ξ ( X X ) ( X X ), π ξ ( X X ) P ( X X ). P P

7 SYMMERIC POSIIE SEMI-DEFINIE 05 herefore, X X mi, if ad oly if W ( X X ) P mi, s. t. W 0. P (9) By Lemma, we coclude that the solutio of the miimizatio roblem (9) is W Wˆ. Substitutio W Wˆ ito (6) yields (8). 4. A Numerical Examle Based o heorems ad, we ca state the followig algorithm. Algorithm. (A algorithm for solvig the otimal aroximatio roblem ()). Iut A, B, C, D, ad X.. If the coditio (5) is satisfied, the we cotiue. Otherwise, we sto. 3. Fid the sigular value decomositios of A ad C accordig to (3) ad (), resectively. 4. If the coditios () ad (4) are satisfied, the the solutio set S X is oemty ad we cotiue. Otherwise, we sto. 5. Comute X 0 ad G 0 by (7) ad (5), resectively. 6. Comute X X G Comute the olar decomositio of the matrix ( X X ) P P XP by (7). J P 8. Comute W ˆ ( J M ). 9. Comute Xˆ accordig to (8).

8 06 Let m, 6, 3. Give A , B , C, D ad , X It is easy to verify that the coditios (5), (), ad (4) hold. Usig Algorithm, we obtai the otimal aroximatio solutio of () as follows. X ˆ It is easily see that Xˆ is a symmetric ositive semi-defiite matrix. Furthermore, we ca figure out A Xˆ B XC ˆ D.5e 0.

9 SYMMERIC POSIIE SEMI-DEFINIE 07 Refereces [] J. P. Aubi, Alied Fuctioal Aalysis, Joh Wiley, New York, 979. [] A. Dajić ad J. J. Koliha, Positive solutios to the equatios AX C ad XB D for Hilbert sace oerators, J. Math. Aal. Al. 333 (007), [3] N. J. Higham, Comutig a earest symmetric ositive semidefiite matrix, Liear Algebra ad its Alicatios 03 (988), [4] F. Li, X. Hu ad L. Zhag, he geeralized reflexive solutio for a class of matrix equatios ( AX B, XC D), Acta Mathematica Scietia 8B (008), [5] Y.-. Li ad W.-J. Wu, Symmetric ad skew-atisymmetric solutios to systems of real quaterio matrix equatios, Comuters ad Mathematics with Alicatios 55 (008), [6] S. K. Mitra, he matrix equatios AX C, XB D, Liear Algebra ad its Alicatios 59 (984), 7-8. [7] Y. Qiu, Z. Zhag ad J. Lu, he matrix equatios with PX sxp costrait, Al. Math. Comut. 89 (007), [8] Y. Qiu ad A. Wag, Least squares solutios to the equatios AX B, XC D with some costraits, Al. Math. Comut. 04 (008), [9] Q.-W. Wag, Bisymmetric ad cetrosymmetric solutios to systems of real quaterio matrix equatios, Comuters ad Mathematics with Alicatios 49 (005), [0] L. Zhag, he solvability coditios for the iverse roblem of symmetric oegative defiite matrices, Mathematica Numerica Siica (989), g