H-matrix theory and applications

MtTrd 205, Combr H-mtrx theory nd pplctons Mj Nedovć Unversty of Nov d, erb jont work wth Ljljn Cvetkovć

Contents! H-mtrces nd DD-property Benefts from H-subclsses! Brekng the DD Addtve nd multplctve condtons Prttonng the ndex set Recursve row sums Nonstrct condtons

H-mtrces nd DD-property A complex mtrx A=[j]nxn s n DDmtrx f for ech from N t holds tht > r A ( ) = j j N, j Deleted row sums Lévy-Desplnques: nonsngulr Lev

H-mtrces nd DD-property A complex mtrx A=[j]nxn s n DDmtrx f for ech from N t holds tht > r A ( ) = j j N, j A complex mtrx A=[j]nxn s n H-mtrx f nd only f there exsts dgonl nonsngulr Lev mtrx W such tht AW s n DD mtrx.

H-mtrces nd DD-property A complex mtrx A=[j]nxn s n DDmtrx f for ech from N t holds tht > r A ( ) = j j N, j H HH DD

H-mtrces nd DD-property A complex mtrx A=[j]nxn s n DDmtrx f for ech from N t holds tht > r A ( ) = j j N, j H HH DD?

ubclsses of H-mtrces & dgonl sclng chrcterztons Benefts:. Nonsngulrty result coverng wder mtrx clss 2. A tghter egenvlue ncluson re (not just for the observed clss) 3. A new bound for the mx-norm of the nverse for wder mtrx clss 4. A tghter bound for the mx-norm of the nverse for some DD mtrces 5. chur complement relted results (closure nd egenvlues) 6. Convergence re for relxton tertve methods 7. ub-drect sums 8. Bounds for determnnts

Brekng the DD Recursve row sums Addtve nd multplctve condtons DD Non-strct condtons Prttonng the ndex set

Ostrowsk, A. M. (937), Pupkov, V. A. (983), Hoffmn, A.J. (2000), Vrg, R.. : Geršgorn nd hs crcles (2004) Brekng the DD Recursve row sums Addtve nd Ostrowsk, multplctve Pupkov condtons DD Non-strct condtons Prttonng the ndex set

Go, Y.M., Xo, H.W. (992), Vrg, R.. (2004), Dshnc, L.., Zusmnovch, M.. (970), Kolotln, l. Yu.(200), Cvetkovć, Lj., Nedovć, M. (2009), (202), (203). Brekng the DD Recursve row sums Addtve nd Ostrowsk, multplctve Pupkov condtons DD Non-strct condtons Prttonng -DD, the ndex PH set

Mehmke, R., Nekrsov, P. (892), Gudkov, V.V. (965), zulc, T. (995), L, W. (998), Cvetkovć, Lj., Kostć, V., Nedovć, M. (204). Brekng the DD Recursve Nekrsovmtrces row sums Addtve nd Ostrowsk, multplctve Pupkov condtons DD Non-strct condtons Prttonng -DD, the ndex PH set

O. Tussky (948), Beuwens (976), zulc,t. (995), L, W. (998), Vrg, R.. (2004) Cvetkovć, Lj., Kostć, V. (2005) Brekng the DD Recursve Nekrsovmtrces row sums Addtve nd Ostrowsk, multplctve Pupkov condtons DD IDD, Non-strct CDD -IDD, condtons -CDD Prttonng -DD, the ndex PH set

O. Tussky (948), Beuwens (976), zulc,t. (995), L, W. (998), Vrg, R.. (2004) Cvetkovć, Lj., Kostć, V. (2005) Brekng the DD Recursve Nekrsovmtrces row sums H Addtve nd Ostrowsk, multplctve Pupkov condtons DD IDD, Non-strct CDD -IDD, condtons -CDD Prttonng -DD, the ndex PH set

I Addtve nd multplctve condtons Ostrowsk-mtrces multplctve condton: jj > r ( A) r ( A) j Pupkov-mtrces ddtve condton: > r ( A) + r ( A) ( A) > mn{mx { }, r } + jj j j j Ostrowsk, A. M. (937), Pupkov, V. A. (983), Hoffmn, A.J. (2000), Vrg, R.. : Geršgorn nd hs crcles (2004)

II Prttonng the ndex set -DD-mtrces Gven ny complex mtrx A=[j]nxn nd gven ny nonempty proper subset of N, A s n -DD mtrx f ( A) =, > r j j, j ( ) > r ( A) r ( A) ( r ( A) ) r ( A), j jj j j, Go, Y.M., Xo, H.W. LAA (992) Cvetkovć, Lj., Kostć, V., Vrg, R. ETNA (2004)

II Prttonng the ndex set! A mtrx A=[j]nxn s n -DD mtrx ff there exsts mtrx W n Ws such tht AW s n DD mtrx. W -DD-mtrces = W = dg( w, w,..., 2 wn ): w = γ > 0 for nd w = for γ.. γ N\.. DD

Dgonl sclng chrcterzton & clng mtrces γ.. γ N\.. DD We choose prmeter from the ntervl: I ( γ ( A) γ ( )), γ = 0, 2 A ( ) mx r ( A) γ A = r ( A), γ 2( A) = j mn jj r j rj ( A) ( A).

Dgonl sclng chrcterzton & clng mtrces -DD DD

Dgonl sclng chrcterzton & clng mtrces γ γ γ -DD T-DD DD γ γ γ γ

Dgonl sclng chrcterzton & clng mtrces γ γ γ -DD -DD T-DD DD γ γ γ γ

Dgonl sclng chrcterzton & clng mtrces γ γ γ -DD DD γ γ γ γ

Egenvlue loclzton ( ) ( ) { },, : A r z C z A = Γ ( ) ( ) ( ) ( ) ( ) ( ) ( ) { }.,, : j A A r r A r z A r z C z A V j j jj j = ( ) ( ) ( ) ( )., Γ = j j A V A A C A σ Cvetkovć, L., Kostć, V., Vrg R..: A new Geršgorn-type egenvlue ncluson set ETNA, 2004. Vrg R..: Geršgorn nd hs crcles, prnger, Berln, 2004.

chur complement A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 chur The chur complement of complex nxn mtrx A, wth respect to proper subset α of ndex set N={, 2,, n}, s denoted by A/ α nd defned to be: A ( α ) A( α, α )( A( α )) A( α, α )

chur complement A ( α) A( α,α ) DD A ( α,α) A( α ) A ( α) A( α,α ) 0 chur Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979), 246-25.

chur complement A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 chur DD Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979), 246-25.

chur complement A ( α) A( α,α ) Ostr A ( α,α) A( α ) A ( α) A( α,α ) 0 chur Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979) L, B., Tstsomeros, M.J. : Doubly dgonlly domnnt mtrces. LAA 26 (997)

chur complement A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 chur Ostr Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979) L, B., Tstsomeros, M.J. : Doubly dgonlly domnnt mtrces. LAA 26 (997)

chur complement A ( α) A( α,α ) $ A ( α,α) A( α ) A ( α) A( α,α ) 0 chur Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979) L, B., Tstsomeros, M.J. : Doubly dgonlly domnnt mtrces. LAA 26 (997) Zhng, F. : The chur complement nd ts pplctons, prnger, NY, (2005).

chur complement A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) $ 0 chur Crlson, D., Mrkhm, T. : chur complements of dgonlly domnnt mtrces. Czech. Mth. J. 29 (04) (979) L, B., Tstsomeros, M.J. : Doubly dgonlly domnnt mtrces. LAA 26 (997) Zhng, F. : The chur complement nd ts pplctons, prnger, NY, (2005).

chur complements of -DD A ( α) A( α,α ) -DD A ( α,α) A( α ) A ( α) A( α,α ) 0 chur Cvetkovć, Lj., Kostć, V., Kovčevć, M., zulc, T. : Further results on H-mtrces nd ther chur complements. AMC (2008) Lu, J., Hung, Y., Zhng, F. : The chur complements of generlzed doubly dgonlly domnnt mtrces. LAA (2004)

chur complements of -DD A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 -DD chur Cvetkovć, Lj., Kostć, V., Kovčevć, M., zulc, T. : Further results on H-mtrces nd ther chur complements. AMC (2008) Lu, J., Hung, Y., Zhng, F. : The chur complements of generlzed doubly dgonlly domnnt mtrces. LAA (2004)

chur complements of -DD Theorem. Let A=[j]nxn be n -DD mtrx. Then for ny nonempty proper subset α of N, A/ α s lso n -DD mtrx. More precsely, f A s n -DD mtrx, then A/ α s n (\ α)-dd mtrx. Theorem2. Let A=[j]nxn be n -DD mtrx. Then for ny nonempty proper subset α of N such tht s subset of α or N\ s subset of α, A/ α s n DD mtrx. Cvetkovć, Lj., Kostć, V., Kovčevć, M., zulc, T. : Further results on H-mtrces nd ther chur complements. AMC (2008) Cvetkovć, Lj., Nedovć, M. : pecl H-mtrces nd ther chur nd dgonl- chur complements. AMC (2009)

Egenvlues of the C ( α) A DD A( α,α ) A ( α,α) A( α ) σ ( A) Γ( A) = Γ ( A) ( A α) λ σ / ') = ( z C : z r A N N *) ( ) = j { } j N \ + ), -) Lu, J., Hung, Z., Zhng, J. : The domnnt degree nd dsc theorem for the chur complement. AMC (200)

Egenvlues of the C of -DD A DD, = α. 50 50 γ > σ R mx α r α ( A) r ( A), α ( A/ α ) = σ ( W AW )/ α ) Γj ( W AW ) j = γr α j α ( A) + r ( A). j j α, 0-50 - 50 0 50 0-50 - 50 0 50 Weghted Geršgorn set for the chur complement mtrx Cvetkovć, Lj., Nedovć, M. : Egenvlue loclzton refnements for the chur complement. AMC (202) Cvetkovć, Lj., Nedovć, M. : Dgonl sclng n egenvlue loclzton for the chur complement. PAMM (203)

Egenvlues of the C! Let A be n DD mtrx wth rel dgonl entres nd let α be proper subset of N. Then, A/α nd A(N\α) hve the sme number of egenvlues whose rel prts re greter (less) thn w (resp. -w), where ( ) r A α w( A) = mn jj rj ( A) + mn rj ( A) j α α A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 C Lu, J., Hung, Z., Zhng, J. : The domnnt degree nd dsc theorem for the chur complement. AMC (200)

Egenvlues of the C of -DD! Let A be n -DD mtrx wth rel dgonl entres nd let α be proper subset of N. Then, A/ α nd A(N\α) hve the sme number of egenvlues whose rel prts re greter (less) thn w (resp. -w), where Remrks: w = ( AW ) w W Ths result covers wder clss of mtrces. By chngng the prmeter n the sclng mtrx we obtn more vertcl bounds wth the sme seprtng property. We cn pply t to n DD mtrx, observng tht t belongs to T-DD clss for ny T subset of N. Cvetkovć, Lj., Nedovć, M. : Egenvlue loclzton refnements for the chur complement. AMC (202)

Egenvlues of the C of -DD A ( α) A( α,α ) A ( α) A( α,α ) A ( α,α) A( α ) 0 C

Dshnc-Zusmnovch A mtrx A=[j]nxn s Dshnc-Zusmnovch (DZ) mtrx f there exsts n ndex n N such tht ( r ( A) + ) > r ( A), j, j N. jj j j j γ.. DD Dshnc, L.., Zusmnovch, M..: O nekotoryh krteryh regulyrnost mtrc loklzc spectr. Zh. vychsl. mtem. mtem. fz. (970) Dshnc, L.., Zusmnovch, M..: K voprosu o loklzc hrkterstcheskh chsel mtrcy. Zh. vychsl. mtem. mtem. fz. (970)

PH-mtrces π : 2,,..., l n x n l x l Aggregted mtrces Kolotln, L. Yu. : Dgonl domnnce chrcterzton of PM- nd PHmtrces. Journl of Mthemtcl cences (200)

PH-mtrces * * * * Kolotln, L. Yu. : Dgonl domnnce chrcterzton of PM- nd PHmtrces. Journl of Mthemtcl cences (200)

Nekrsov mtrces! A complex mtrx A=[j]nxn s DDmtrx f for ech from N t holds tht ( A) r( A) d > > r ( A) = j j N, j! A complex mtrx A=[j]nxn s Nekrsov-mtrx f for ech from N t holds tht ( A) h( A) d > h > h ( A) = ( A), h ( A) = r ( A) h ( A) n j= j j jj + j= + j,, = 2,3,..., n. A = D L U

Nekrsov mtrces A complex mtrx A=[j]nxn s Nekrsov-mtrx f for ech from N t holds tht h > ( A) h = ( A), h ( A) = r ( A) h ( A) n j= j = 2,3,..., n. j ( A) h( A) jj + j= + Nekrsov row sums j,, d > A = D L U

Nekrsov mtrces nd sclng Theorem. Let A=[j]nxn be Nekrsov mtrx wth nonzero Nekrsov row sums. Then, for dgonl postve mtrx D where d h = ε ( A), =,, n, nd ( ε ) n = s n ncresng sequence of numbers wth ε =, ε,, = 2,, n, h ( A) the mtrx AD s n DD mtrx. zulc, T., Cvetkovć, Lj., Nedovć, M. : clng technque for Nekrsov mtrces. AMC (205) (n prnt)

Nekrsov mtrces nd permuttons! Unlke DD nd H, Nekrsov clss s NOT closed under smlrty (smultneous) permuttons of rows nd columns!! Gven permutton mtrx P, complex mtrx A=[j]nxn s clled P-Nekrsov T T ( P AP) > h ( P AP) T T ( P AP) > h( P AP). d f, N, The unon of ll P-Nekrsov= Gudkov clss A = D L U

{P,P2} Nekrsov mtrces! uppose tht for the gven mtrx A=[j]nxn nd two gven permutton mtrces P nd P2 d { } ( ) ( ) P T, h A = P h P AP,,2. P P2 ( A) mn h ( A), h ( A) > k k = We cll such mtrx {P,P2} Nekrsov mtrx. k k k A A A2

{P,P2} Nekrsov mtrces o Theorem. Every {P, P2} Nekrsov mtrx s nonsngulr. o Theorem2. Every {P, P2} Nekrsov mtrx s n H mtrx. o Theorem3. Gven n rbtrry set of permutton mtrces Π = { } p k k n P = every Пn Nekrsov mtrx s nonsngulr, moreover, t s n H mtrx. Cvetkovć, Lj., Kostć, V., Nedovć, M. : Generlztons of Nekrsov mtrces nd pplctons. (204)

Mx-norm bounds for the nverse of {P,P2} Nekrsov mtrces o Theorem. uppose tht for gven set of permutton mtrces {P, P2}, complex mtrx A=[j]nxn, n>, s {P, P2} Nekrsov mtrx. Then, A P z mx mn N P h mn mn N P2 ( A) z ( A), P2 ( A) h ( A),, where z z = = j= ( A) r ( A), z ( A) ( A) +, = 2,3,..., n, T P T ( A) = [ z ( A),, z ( A) ], z ( A) = Pz( P AP). n j z j jj Cvetkovć, Lj., Kostć, V., Nedovć, M. : Generlztons of Nekrsov mtrces nd pplctons. (204)

Mx-norm bounds for the nverse of {P,P2} Nekrsov mtrces o Theorem2. uppose tht for gven set of permutton mtrces {P, P2}, complex mtrx A=[j]nxn, n>, s {P, P2} Nekrsov mtrx. Then, A mx N mn% N & { } { ( )} % P mn z P ( A), z 2 & ( A) ' ( P mn h P ( A), h 2 A, ' ( z z = = j= ( A) r ( A), z ( A) ( A) +, = 2,3,..., n, T P T ( A) = [ z ( A),, z ( A) ], z ( A) = Pz( P AP). n Cvetkovć, Lj., Kostć, V., Nedovć, M. : Generlztons of Nekrsov mtrces nd pplctons. (204) j z j jj

Numercl exmples o Observe the gven mtrx B nd permutton mtrces P nd P2. B = 60 75 60 5 5 05 60 5 5 45 20 5 5 0, 5 45 P = 0 0 0 0 0 0 0 0 0 0 0, 0 P 2 = I, h h P P 2 P P P ( B) = 45, h2 ( B) = 0.25, h3 ( B) = 6.607, h4 ( B) = 4.25, P2 P2 P2 ( B) = 45, h ( B) = 0.25, h ( B) = 7.857, h ( B) = 40.4464. 2 3 4 o Notce tht B s Nekrsov mtrx. Cvetkovć, Lj., Kostć, V., Doroslovčk, K. : Mx-norm bounds for the nverse of - Nekrsov mtrces. (202)

Numercl exmples o Observe the gven mtrx B nd permutton mtrces P nd P2. B = 60 75 60 5 5 05 60 5 5 45 20 5 5 0, 5 45 P = 0 0 0 0 0 0 0 0 0 0 0, 0 P 2 = I, B B B.76842 0.96842 = 0.6843. Although the gven mtrx I Nekrsov mtrx, n ths wy we obtned better bound for the norm of the nverse.

Nekrsov mtrces Gven ny mtrx A nd ny nonempty proper subset of N we sy tht A s n -Nekrsov mtrx f jj > > h h j ( A),, ( A), j, ( h ( A) ) h ( A) ( ) > h ( A) h ( A),, j. jj j h h j ( A) r ( A) = ( A) = j= j h j ( A) jj + n j j= +, j If there exsts nonempty proper subset of N such tht A s n - Nekrsov mtrx, then we sy tht A belongs to the clss of -Nekrsov mtrces.

Nekrsov mtrces Cvetkovć, Lj., Kostć, V., Rušk,. : A new subclss of H- mtrces. AMC (2009) W = W = dg( w, w,..., 2 wn ): w = γ > 0 for nd w = for γ.. γ N\.. Nekrsov

Nekrsov mtrces Cvetkovć, Lj., Kostć, V., Rušk,. : A new subclss of H- mtrces. AMC (2009) Cvetkovć, Lj., Nedovć, M. : pecl H-mtrces nd ther chur nd dgonl-chur complements. AMC (2009) zulc, T., Cvetkovć, Lj., Nedovć, M. : clng technque for Nekrsov mtrces. AMC (205) (n prnt)

Nonstrct condtons DD - mtrces IDD-mtrces ( A),,2,..., n. r = kk > r k ( A) for one k n N CDD-mtrces ( A),,2,..., n. r = r ( A) kk > k for one k n N rreducblty non-zero chns Olg Tussky (948) T. zulc (995)

Nonstrct condtons -IDD Gven n rreducble complex mtrx A=[j]nxn, f there s nonempty proper subset of N such tht the followng condtons hold, where the lst nequlty becomes strct for t lest one pr of ndces n nd j n N\, then A s n H-mtrx. r ( A) =, j j, j ( ) r ( A) r ( A),, j. ( r ( A ) r ( A) jj j Cvetkovć, Lj., Kostć, V. : New crter for dentfyng H-mtrces. JCAM (2005) j

Nonstrct condtons -CDD Gven complex mtrx A=[j]nxn, f there s nonempty proper subset of N such tht the followng condtons hold, where the lst nequlty becomes strct for t lest one pr of ndces n nd j n N\, nd for every pr of ndces n nd j n N\ for whch equlty holds there exsts pr of ndces k n nd l n N\ for whch strct nequlty holds nd there s pth from to l nd from j to k, then A s n H-mtrx. r ( A) =, j j, j ( ) r ( A) r ( A),, j. ( r ( A ) r ( A) jj j Cvetkovć, Lj., Kostć, V. : New crter for dentfyng H-mtrces. JCAM (2005) j

Cvetkovć, Lj., Kostć, V., Kovčevć, M., zulc, T. : Further results on H-mtrces nd ther chur complements. AMC (2008) Cvetkovć, Lj., Nedovć, M. : pecl H-mtrces nd ther chur nd dgonl- chur complements. AMC (2009) Cvetkovć, Lj., Nedovć, M. : Egenvlue loclzton refnements for the chur complement. AMC (202) Cvetkovć, Lj., Nedovć, M. : Dgonl sclng n egenvlue loclzton for the chur complement. PAMM (203) Cvetkovć, Lj., Kostć, V., Nedovć, M. : Generlztons of Nekrsov mtrces nd pplctons. (204) zulc, T., Cvetkovć, Lj., Nedovć, M. : clng technque for Nekrsov mtrces. AMC (205) (n prnt)

Mt Trd Combr 205 THANK YOU FOR YOUR ATTENTION! HVALA NA PAŽNJI!