Several Theorems for the Trace of Self-conjugate Quaternion Matrix

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1 Moder Aled Scece Setember, 008 Several Theorems for the Trace of Self-cojugate Quatero Matrx Qglog Hu Deartmet of Egeerg Techology Xchag College Xchag, Schua, 6503, Cha E-mal: Lm Zou(Corresodg author College of Mathematcs ad Physcs Chogqg Uversty Chogqg, , Cha E-mal: The research s Suorted by Chogqg Uversty ostgraduates Scece ad Iovato Fud (0080A A Abstract The urose of ths aer s to dscuss the equaltes for the trace of self-cojugate quatero matrx We reset the equalty for egevalues ad trace of self-cojugate quatero matrces Based o the equalty above, we obta several equaltes for the trace of quatero ostve defte matrx Keywords: Quatero matrx, Trace, Iequaltes, Egevalues Itroducto Quatero was troduced by the Irsh mathematca Hamlto ( The lterature o quatero matrces, though datg bac to 936 [], s fragmetary Quatero s mostly used comuter vso because of ther ablty to rereset rotatos 4D saces It s also used rogrammg vdeo games ad cotrollg sacecrafts [, 3] ad so forth The research o mathematcal objects assocated wth quatero has bee dyamc for years There are may research aers ublshed a varety of jourals each year ad dfferet aroaches have bee tae for dfferet uroses, ad the study of quatero matrces s stll develomet As s exected, the ma obstacle the study of quatero matrces s the o-commutatve multlcato of quatero The theory o egevalues, determat, sgular values ad trace of real ad comlex matrces has bee well establshed O the cotrary, lttle s ow for the trace of quatero matrces As usual, R ad C are the set of the real ad comlex umbers We deote by H ( hoor of the vetor, Hamlto the set of real quatero: {,,,, } H = a= a + a+ a j+ a a a a a R For a = a0 + a + aj+ a3 H, the cojugate of a s a= a = a0 a aj a3 ad the orm of a s / N( a = aa = aa = ( a0 + a + a + a3 Let H ad H be resectvely the collectos of all matrces wth etres H ad -colum vectors Let I be the collectos of all ut matrces wth etres H For X H T, X s the trasose of X If X ( X, X X T = K, the X = ( X, X, K, X T s the cojugate of X ad X = ( X, X, K, X s the cojugate trasose of X The orm of X s defed to be N ( X = X X For a

2 Vol, No 5 Moder Aled Scece matrx A = ( aj ( aj H, the cojugate trasose of A s the matrx T A = A = ( a j The research of matrces s cotuously a mortat asect of algebra roblems over quatero dvso algebra, the subject, such as egevalues, sgular values, cogruet ad ostve defte of self- cojugate matrces as well as sub-determat of self- cojugate matrces ad so o, has bee extesvely exlored [4-5], whle lttle s ow for the trace of quatero matrces For lear algebrasts ad matrx theores, some basc questos o the trace of quatero matrx are dfferet from real or comlex matrx For stace, f A ad B are the matrces, the Tr ( AB = Tr ( BA ad Tr ( A = λ are ot always rght I ths secto, we troduce the otato ad termology I secto, we defe some deftos ad recall several lemmas I secto 3, we dscuss the equalty about egevalues ad trace of self-cojugate quatero matrces I secto 4, we coclude the aer wth several equaltes for the trace of quatero ostve defte matrx obtaed by the result secto 3 ad the Holder s equalty over quatero dvso algebra Deftos ad Lemmas We beg ths secto wth some basc deftos ad lemmas Defto Let A H A s sad to be the self-cojugate quatero matrx f A = A H (, s the set of self-cojugate quatero matrces, H(, > s the set of quatero ostve defte matrces Defto Let A H A s sad to be the quatero utary matrx f A A = AA = I H, u s the set of quatero utary matrces a = Defto 3 Let A H Lemma [4] Let A H (, U H(, where, s sad to be the trace of matrx A, remared by Tr ( A That s Tr( A = a = The, A s utary smlar to a real dagoal matrx, that s, there exsts a utary matrx ( λ λ L, λ U AU = dag,,, λ, λ, Lλ R are the egevalues of A Lemma Let A H (, ad B H(, If there exsts U H(, Proof Sce A H (,, by Lemma, there exsts V H(, where, λ, λ, Lλ R B = UAU, the, TrA = TrB V AV dag λ, λ,, λ = L, are the egevalues of A For ay U H(,, we have uu j j = N ( uj =, ( j=,, L, ( = = j j j j= j= uu = N ( u =, ( =,, L, (3 So TrA = a = v λδ l v = λ l vδlv l = = = l= = = l= = λlvv ll = ll λl vv ll = λ ll l = l= = l= = Meawhle, we have B = UAU, the B = UVdag ( λ, λ, L, λ V U Sce UV H (, TrA TrB λ Thus = = = l, therefore

3 Moder Aled Scece Setember, 008 TrA = TrB (4 3 The equalty for the trace of self-cojugate quatero matrces It s well ow that the egevalues ad trace of ay self-cojugated quatero matrx are all real umbers I ths secto, we shall dscuss the equalty about egevalues ad trace of self-cojugate quatero matrces Theorem Let A H(,, B H(, >, ther egevalues are α α L α, β β L β resectvely, f A, B are commutatve, the Tr ( AB α β = Proof Sce A H(,, B H(,, by lemma, there exsts utary matrces U, U H(, where α > 0 ( =,, L By (3 ad (3, we have Let UU U ( uj ( α, α,, α ( β β, β U AU = dag L (3 U BU dag,, = L (3 Therefore ( α, α,, α ( β, β,, β Tr U dag L U U dag L U = ( α, α,, α ( β, β,, β U ABU = dag L U U dag L U U = =, t s easy to ow U H(,, the uu j j = N ( uj =, ( j=,, L, = = j j j j= = uu = N ( u =, ( =,, L, Sce AB = B A = BA, ad A, B are commutatve, the ( AB Tr ( UABU =, the ( α, α, L, α ( β, β, L, β = Tr dag Udag U = AB, so AB H(, Hece, by Lemma, Let the For ay ( <, we have = ( α, α, L, α N ( N ( u L N ( u β N ( u N ( u L N ( u β L L L L M N ( u N ( u N ( β L ξ N ( N ( u L N ( u β ξ N ( u N ( u L N ( u β = M L L L L M ξ N ( u N ( u N ( β L ξ = β (33 = = ξ β β β β = N ( u = N( u + N ( u j j j j j = = j= = = j= = j= + + β β N ( uj β N ( uj = = j= = j= + = β β N ( uj + β N ( uj = = j= j= 3

4 Vol, No 5 Moder Aled Scece = β (34 = By (33, (34,ad α > 0 ( =,, L, the = α ξ α β = = 4 The equalty for the trace of quatero ostve defte matrx I ths secto, we frst obta a equalty for the trace of two quatero ostve defte matrces based o Theorem The by Theorem ad the Holder s equalty over quatero dvso algebra, the equalty for trace of the sum ad multlcato of quatero ostve defte matrces s obtaed Theorem Let A H(,, B H(, A, B are commutatve, the Proof Sce > >, ther egevalues are α α L α, β β L β resectvely, f Tr ( A Tr ( B Tr A = a = α > ad = =, by Theorem,we have = = Tr B = b = β > 0 Because of A H(, >, the there exsts U H(, where α > 0 ( =,, L Sce B H(, So, α β = U AU = dag α α L α,,, ( α α α U ABU = U AUU BU = dag,, L, U BU (4 >, the B ad I are self-cogruet, hece U BU ad I are self-cogruet, that s, U BUs quatero ostve defte matrx For ay ma sub-matrx L of U ABU, by (4, we ow that L ca be obtaed by the ma sub-matrx G of U BUThe row row L = L = αα L G = αα L G > 0, j j where s the determat of quatero matrx defed by Che L X [0] So, U ABU s quatero ostve Sce defte matrx, hece Tr ( AB > 0 Tr( A Tr( B α β α β 0 so Because of the amely so = = = Tr( A Tr( B ( Tr A Tr B Tr A Tr A Tr B Tr B (43 4

5 Moder Aled Scece Setember, 008 By (4 ad (43, the cocluso holds Theorem 3 Let A H(,, B H(, > > Ther egevalues are α α L α, β β L β resectvely, f >, + q =,ad A, B are commutatve, the Proof By Theorem ad the Holder s equalty, we have q ( ( q Tr A Tr B = = = Tr A Tr B Secally, whe = q=, we have Theorem 4 Let A H(,, B H(, >,ad A, B are commutatve, the Proof Let q q P q ( ( q αβ α β = ( ( Tr A Tr B > > Ther egevalues are α α L α, β β L β resectvely, f ( ( q =, the, q 0, > > + q = Sce the, by Theorem 3 ad the Holder s equalty, we have ( Tr A + B Tr A + Tr B ( A+ B = A( A+ B + B( A+ B ( + = ( + + ( + Tr A B Tr A A B B A B q q P q q P ( TrA Tr A + B + TrB Tr A + B ( q q P P = Tr A + B TrA + TrB That s So P P = Tr (( A + B ( TrA + ( TrB ( P P = Tr A + B TrA + TrB P P Tr A + B Tr A + B TrA + TrB ( ( ( Tr A + B Tr A + Tr B Refereces Che Logxua Defto of determat ad Cramer solutos over the quatero Acta Math Sca, New ser99 ( Huag Lg, Wa Zhexa Geometry of sew-hermta matrces [J] Lear Algebra Al 396( Huag Lg O two questos about quatero matrces [J] L Alg Al 33( LA Wolf, Smlarty of matrces whch elemets s real quatero [J] Bull Amer Math Soc 4 (

6 Vol, No 5 Moder Aled Scece Ra Rusheg, Huag Tgzhu The recogto of color Images based o the sgular value decomostos of quatero matrces [J] ( Chese Comuter Scece 006(077-9 Tu Boxue Wea roecer roduct ad wea Hadamard roduct of Quatero matrx [J] ( Chese Joural of Fu da uv 99 ( Tu Boxu The cetralzed basc theorem of a real quatero self-cojugate matrx ad ts alcato [J] ( Chese Joural of Mathematcs, 988(8, Wag Qggu Quatero trasformato ad ts alcato o the dslay-cemet aalyss of satal mechasms [J] ( Chese Acta Mechaca Sca983(0, 54-6 Wag Qgwe, Sog Guagjg, L Chuya Extreme ras of the soluto to a cosstet system of lear quatero matrx equatos wth a alcato [J] Al Math Com89 ( Wag Qgwe A system of matrx equato ad a lear matrx equato over arbtrary regular rgs wth detty[j] Lear Algebra Al 384 ( Wag Qgwe The Geeral Soluto to a system of real quatero matrx equato [J] Comuter ad Mathematcs wth Alcatos 49 ( Wu Julag Dstrbuto ad Estmato for Egevalues of Real Quatero Matrces [J] Comuters ad Mathematcs wth Alcatos 55 ( Zhag Fuzhe Quateros ad matrces of quateros [J] Lear Algebra Al 5 ( Zhag Fuzhe Gersgor tye theorems for quateroc matrces [J] Lear Algebra Al 44 ( Zhuag Waj The gude of quatero algebra theory [M] ( Chese Scece ress, Bejg, 006 6