bstrat.4 6....3 9 8 3
. B, α,,α ( ρ) ρee, β,, β ρ, C = ρee ( ρ) B, ρ,( ρ) α,,( ρ) α,( ρ) β,,( ρ) β., α,, α },, β,, β },, α,, α, β,, β} 3. σ =, λ,, λ } B λ,,λ S R BS S σ,,,, λ, µ ( λ < µ ) x, y x, y
( λ, x) ( µ, y) λ, µ ( λ < µ ) x, y ) ( λ, x ( µ, y) 3
bstrat he verse egevalue problems for matres are studed may felds. hey arse a remarkable varety of applatos, whh ot oly lude dspersed mathematal physal verse problems, but sold physs, partle physs, quatum mehas, struture aalyss, otrol desg, system parameter detfato, stohast proess, ad so o. here are also some verse egevalue problems for matres umeral algebra. he paper maly studes verse egevalue problems for symmetr doubly stohast matres ad doubly symmetr fvedagoal matres. For symmetr doubly stohast matres, we have some results as follows:. Let B be symmetr doubly stohast matres wth respetve egevalues matrx C = ( ρ) ρee, α,,α,, β,, β ρee ( ρ) B,the ρ,the s symmetr doubly stohast matrx, ad ts egevalues are, ρ, ρ) α,, ρ) α, ( ρ) β,, ( ρ) β. ( (. Suppose, α,, α },, β,, β } both a be realzed by symmetr doubly stohast matrx, the,, α,, α, β,, β } a be realzed by symmetr doubly stohast matrx. 4
3. Suppose σ =, λ,, λ }, f there exsts some real symmetr matrx B wth egevalues λ,,λ, ad some smplex S R wth ts vertes are orthogoal eah other ad the ozero vertes are ut vetors suh that BS S, the σ a be realzed by olum stohast matrx whose the th prpal submatrx s symmetr ad the last olum, s elemets are,,,,. For doubly symmetr fve-dagoal matres, we maly resolve two problems as follows: Problem : Gve two dstt real salars λ, µ ( λ < µ ) ad two ozero real symmetr vetors symmetr vetors matres suh that x, y R or two ozero real at- x, y R, fd doubly symmetr fve-dagoal x = λx ad y = µ y. Problem : Gve two dstt real salars λ, µ ( λ < µ ) ad two ozero real vetors at-symmetr, fd suh that x, y R x = λx ad y = µ y., oe s symmetr, the other s doubly symmetr fve-dagoal matres Keywords: verse egevalue problems stohast matres doubly symmetr matres 5
λ,,λ x R x = x m = Μ Μ m m Μ m m Μ mm 6 = dag λi [ ],, [ m ] λ I } m [ ] I x R x ( x,, x ) = x R m x =, =,, m m = [] [] [3] [4]
= ( a j ) m a j, =,,, m, j =,,, j = = ( a j ) m a j = s, =,,, m 3 = ( a j ) a j =, =,,, j = a =, j =,, = aj =, aj =,, j =,,, = j = j, e = (,,,), J = ee J e = e J = J e = e J = J, J = J a J = J e = e, e = e m 4 a, a,, a R m a a, a a,, a x = = λ a, λ =, λ, =,,, = S R m S = ( a, a,, a) a, a,, a S S = ( a, a,, a) x = λa = 7
λ, λ,, λ ) S ( 5 P P P =, 6 = ( a j ) aj = a j, aj = a+, + j,, j =,,, S = ( e, e,, e ) e I =, S S = 7 z = ( z, z,, z ) z = z + z z = z + z S z S z k z z z = z, z,, ) = k z k + k z ( z k z = k + z Sk z z S k = ( ek, ek,, e ) Sk z a 8 b = Jaob b a Ο Ο Ο b b a b >, =,,, 9 C 8
9 C = 3 3 C ( ) = C ) ( r ),,, ( =
. σ = λ, λ,, λ } σ ( ) σ = λ, λ,, λ } C ( ) σ = λ, λ,, λ } R ( 3) σ = λ, λ,, λ } R
( 4) σ = λ, λ,, λ } C ( 5) σ = λ, λ,, λ } R ( 6) σ = λ, λ,, λ } R ( C ) ( 7) σ = λ, λ,, λ } R ( C ) ( 8) σ = λ, λ,, λ } R ( ) ) ( 4 : λ x = x, x,, ) D = dag x, x,, ( x ( x ) De = λ e e = (,, ) D D D P P P = * * Ο * k D D λ λ = max( λ ) j j D D ~ λ λ j j =,, k = D Μ k D Μ Ο D k D k k
λ [] x = x, x,, ) D D ( x D = dag x, x,, x ) D D ( [9] σ = λ =, λ,, λ } ρ ( ) ρ ( ) λ λ + λ + λ3 + + ( ) ( )( ) λ = V DV V = D = dag( λ, λ,, λ ) V ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) 6 6 6 V D () Loewy Lodo [5] 3 = = 4 5 Reams[6] Laffey Meeha[7]
Rojo Soto [7] [] [] [4] [5] [6] [7] [8] [9] [] [] [] [3] [4].. e = e, e = e J = J, J = J. 3. 4. 5. t = P P > =,, t = = 6. = ( a j ) x = ( x,, ) λ λ x = = x t = x = λx x = a a a 3 x + x + x + + a + a + a x x x x
x a + + xa = x + + x = λ( x + + x) λ = = x = = [9] 7. r r r > r r r =,,r r r [ ]. σ, λ,, λ }, λ λ = ) + λ + λ3 + + λ ( ) ( )( ) δ δ δ3 δ ) + + + + + δ 3 ) h 4 ) ( ) h σ δ λ λ =,,, δ =, = + = (, λ,, λ = (, δ,, δ ), h = (,,,, ), 3 ),, ) a b a b λ δ h = (, 3, ( ) = [ ]. σ =, λ,, λ}, λ,, λ [,] σ ( ) 4
+ m k = [ 3] m 3. σ =, λ,, λ} λ λ + λ ( m + ) λ k + ( k + ) k [ 9] m = σ + 4. σ =, λ,, λ, }, k Z, = k σ k σ, λ,, λ, λ,, λ, } = k k [ 4] 5. 3 σ =,,,,,,} m m α, α,, αm, u = αu u = ; B β, β,, β Bv = βv v = C = ρvu ρuv B ρ λ, λ, α,, α m, β, β λ,λ ^ α ρ C = ρ β B, α,,α ( ρ) ρee, β,, β ρ, C = ρee ( ρ) B, ρ,( ρ) α,,( ρ) α,( ρ) β,,( ρ) β B C e, u,, u u = α u =,, 5
e, v,, v u j j B Bv = β v j =,, C ( ρ) α C ( ρ) β v =,, r s r s ^ C = ρ ρ j ρ ρ λ = ρ ρ r r λ = ρ = λ =, ρ ρ s s re s e C C λ =, r e s e = ( ρ) ρee ( ρ) re + ρs e ( ρ) s e + ρre ρee ( ρ) B = λ re s e re s e = =, ( ρ) r e + ρs ee e ρree e + ( ρ) s Be C, ρ, ρ) α, ( ρ) α, ( ρ) β,,( ρ) β (,, α,, α },, β,, β },, α,, α, β,, β} ρ = [] Boroba σ =, λ,, λ } B λ,,λ S R S BS S 6
Degree papers are the Xame Uversty Eletro heses ad Dssertatos Database. Full texts are avalable the followg ways:. If your lbrary s a CLIS member lbrares, please log o http://etd.als.edu./ ad submt requests ole, or osult the terlbrary loa departmet your lbrary.. For users of o-clis member lbrares, please mal to etd@xmu.edu. for delvery detals.