Stability of (A, B)-invariant subspaces

Transcription

1 Uiversitat Politècica e Cataluya Escola Tècica Superior Egiyeria Iustrial e Barceloa Stability of (A, B)-ivariat subspaces Marta Peña, Ferra Puerta, Xavier Puerta Departamet e Matemàtica Aplicaa I Diagoal 647, Barceloa, Spai ETSEIB, UPC Regia 005

2

3 Cotext Characterizatio of stable ivariat subspaces of a eomorphism: oe of a pair of matrices: ope problem 1

4 Notatio { M = ( ABW,,, F): A M ( F), B M ( F), W Gr ( F ), N, m F M ( F),( A+ BF) W W m, = M ( F) M ( F) Gr ( F ) M ( F), m m, } Gr ( F ) set of -imesioal subspaces of F F= or P set of orthogoal projector operators of rak, P { P M ( F): P * P, P P, rakp } = = = =

5 Aim To obtai computable coitios of stability of (A,B)-ivariat subspaces Local cooriate charts of M a N M is a maifol Sufficiet coitio of stability 3

6 Eomorphisms A-ivariat subspace Defiitio A M ( F ), W Gr ( F ) A( W) W A-stable ivariat subspace Defiitio A M ( F ), W Gr ( F ) ivariat is stable if give ε > 0 there exists a δ > 0 such that A' A < δ for a liear map A' M ( F ) implies that A ' has a ivariat subspace W ' with Θ ( WW, ') < ε 4

7 Pairs of matrices (A,B)-ivariat subspace Defiitio A M ( F ), B M m, ( F ), W Gr ( F ) A( W) W + ImB (A,B)-stable ivariat subspace Defiitio A M ( F ), B M m, ( F ), W Gr ( F ) (A,B)-ivariat is (A,B)-stable if give ε > 0 there exists a δ > 0 such that ( A', B') ( A, B) < δ for a pair A' M ( F ), B' M m, ( F ) implies that ( A', B ') has a ( A', B') -ivariat subspace W ' Θ ( WW, ') < ε with 5

8 Differetiable structure of M Suggestio of U. Helmke ( P ) Prelimiary results Propositio (i) P submaifol of M ( F ) im P = ( ) (ii) { * T [, ]:, ( )} PP = P Ω Ω= Ω Ω M F (iii) P Gr ( F ) [ P, Ω ] = PΩ Ω P Propositio N = M ( F) M ( F) P M ( F ), m m, {( A, BPF,, ) N : ( A BFP ) PA ( BFP ) } M = + = + 6

9 Differetiable structure of M Local cooriate charts of M a N We cosier ( A0, B0, W0, F0) N such that I W0 = Im 0 Lemma N 0 { ABW F A M F B M, m = (,,, ) : ( ), ( F), I W = Im, Q M, ( ), F Mm, ( ) Q F F is a ope set of N that cotais ( A0, B0, W0, F 0) Lemma M 0 A A B I,,Im,[ F F ] : 1 1 = 1 A3 A 4 B Q A = QA A Q + QA Q + QB F + QB F Q B F B F Q } 7

10 Differetiable structure of M Local cooriate charts of M a N Propositio γ: F + ( ) + m N 0 γ ( A, A, A, A, B, B, Q, F, F ) = A A B I,,Im, [ F F ] 1 1 = 1 A3 A 4 B Q ( N 0,γ) cooriate system of the maifol N Theorem ψ : F F + m + ( ) + m ψ ( A1, A, A4, B1, B, Q, F1, F) = ( A1, A, QA1 AQ 4 + QAQ + + QB F + QB F Q B F B F Q, A, B, B, Q, F, F ) θ:=γ ψ ( M 0,θ) cooriate system of M M is a maifol, imm = + m 8

11 Differetiable structure of M F + ( ) + m γ A1 A B1 I ( A1, A, A3, A4, B1, B, Q, F1, F),,Im,[ F1 F] A3 A 4 B Q ( A1, A, QA1 AQ 4 + QAQ + QBF 1 1+ QB1F Q BF1 B FQ, A4, B1, B, Q, F1, F) M0 ψ θ = γ ψ N 0 F ( A1, A, A4, B1, B, Q, F1, F) + m Theorem M submaifol of N Propositio χ = ( A, BPF,, ) M, { T χ M= ( ABPF,,, ): A M ( F), B M, m( F), P TPP, F M m, ( F), ( I P)( AP + AP + BFP + BFP + BFP ) P ( A + BF) P= 0 } 9

12 Differetiable structure of M Proof (Taget space of M) Smooth map ϕ ϕ : N ( F) M χ = ( A, BPF,, ) ( A+ BFP ) PA ( + BFP ) M 1 =ϕ (0) ϕ ( A, B, P, F χ ) = ( A + BF + BF ) P+ ( A+ BF) P P ( A + BF) P P( A + BF + BF ) P P( A + BF) P * * X, L = tr( XL) ; L ( Im χ ) ( ϕ tr ( A + BF + BF ) P+ ( A+ BF) P P ( A+ BF) P+ + [ P, Ω ]( L( A+ BF) LP( A+ BF) ( A+ BF) PL) = 0 A M ( F), B M ( F), F M ( F),, m m, * Ω M ( F), Ω = Ω ) 10

13 Differetiable structure of M PL( I P) = 0 ( ) { * Im ( ) : ( ) 0 } χ L M F PL I P ϕ = = = { L M ( F): ( I P) LP 0} = = L 3 = 0 I 0 P = 0 0, L L L 1 = L3 L 4 rak ϕ ( ) χ = im M= im N rak ϕ = χ = ( + ( ) + m) ( ) = + m ( ϕ χ) = im( TχM ) im Ker T χ M = Ker ϕ χ 11

14 Stability of (A,B)-ivariat subspaces Properties of π M ( F) M ( F) M ( F), m m, π π 1 1 M N π π Gr ( F ) π 1 ( A, BPF,, ) = ( ABF,, ) π ( A, BPF,, ) = ImP Propositio (i) χ = ( A, BPF,, ) M, rak π, χ = ( ) (ii) π : M Gr ( F ) is a submersio 1

15 Stability of (A,B)-ivariat subspaces Sufficiet coitio of stability Theorem χ = ( A, BPF,, ) M, π bijective W ( A, B)-stable 1,χ { P TPP :( I P)( A+ BF) P P ( A+ BF) P= 0} = {} 0 13

16 . Compariso with stability of A-ivariat subspaces Give a eomorphism, { ( A, W): A M ( ), Gr ( ), ( ) } F W F A W W M = N = M ( F) Gr ( F ) M ( F) π π 1 1 M N π π Gr ( F ) π 1 ( A, W) = A π ( A, W) = W Sufficiet coitio of stability Theorem ( A, W ) M, π bijective W A stable 1, ( AW, ) PΩ AP+ΩAP AΩ P+ PAΩ P= 0 ( ), P P * Ω=Ω Ω M F Ω= Ω 14

17 Refereces [1] A. Compta; U. Helmke; M. Peña; X. Puerta, Simultaeous Versal Deformatios of Eomorphisms a Ivariat Subspaces, Liear Algebra Appl. [] I. Gohberg; P. Lacaster; L. Roma, Ivariat subspaces of Matrices with Applicatios, Wiley, New York (1986). [3] L. Roma, Stable Ivariat Subspaces Moulo a Subspace, Operator Theory, Avaces a Applicatios, vol. 19, , Birkhauser Verlag Bassel (1986). [4] F. Velasco, Stable Subspaces of Matrix Pairs, Liear Algebra Appl., 301 (1999), p [5] W. Woham, Liear Multivariable Cotrol: A Geometric Approach, Spriger, New York (1979). 15