On singular two-parameter eigenvalue problems Andrej Muhi c - PowerPoint PPT Presentation

MEP From MEP to a coupled GEP Common regular eigenvector On singular two-parameter eigenvalue problems Andrej Muhiˇ c Institute of Mathematics, Physics and Mechanics, University of Ljubljana, Slovenia The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, May 10-17, 2009 This is joint work with B. Plestenjak. The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Outline Two parameter eigenvalue problem 1 From MEP to a coupled GEP 2 From common regular eigenvector to eigenvalue of MEP 3 The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Two-parameter eigenvalue problem Two-parameter eigenvalue problem: W 1 ( λ, µ ) x 1 := ( A 1 + λ B 1 + µ C 1 ) x 1 = 0 ( MEP ) W 2 ( λ, µ ) x 2 := ( A 2 + λ B 2 + µ C 2 ) x 2 = 0 Eigenvalue: a pair ( λ, µ ) satisfying (MEP) for nonzero x 1 and x 2 . Eigenvector: the tensor product x 1 ⊗ x 2 . Equivalent problem : finding common zeros of polynomials p 1 ( λ, µ ) = det ( W 1 ( λ, µ )) and p 2 ( λ, µ ) = det ( W 2 ( λ, µ )) . The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Tensor product approach ∆ i matrices on the space C n 1 × n 2 ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 ∆ 1 = C 1 ⊗ A 2 − A 1 ⊗ C 2 ∆ 2 = A 1 ⊗ B 2 − B 1 ⊗ A 2 . MEP is nonsingular ⇐ ⇒ some combination of ∆ i (usually ∆ 0 ) is nonsingular. MEP is eiquivalent to a coupled pair of generalized eigenvalue problems ∆ 1 z = λ ∆ 0 z ( GEP ) ∆ 2 z = µ ∆ 0 z , where z = x 1 ⊗ x 2 . ∆ − 1 0 ∆ 1 and ∆ − 1 0 ∆ 2 commute. The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector The singular two-parameter eigenvalue problem every combination a ∆ 0 + b ∆ 1 + c ∆ 2 is singular pencils λ ∆ 0 − ∆ 1 and µ ∆ 0 − ∆ 2 are singular Model updating (Cottin 2001, Cottin and Reetz 2006): finite element models of multibody systems are updated to match the measured input-output data. Spectrum of delay-differential equations (Jahrlebring 2008) Linearization of QMEP (Muhiˇ c, Plestenjak 2008) the numerical method for the solution of the singular MEP The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Some definitions eigenvalue ω is a finite regular eigenvalue of matrix pencil λ B − A if and only if rank ( ω B − A ) < max s ∈ C rank ( sB − A ) = n r . Normal rank of W i ( λ, µ ) : nrank ( W i ( λ, µ )) = max λ,µ ∈ C rank ( W i ( λ, µ )) , i = 1 , 2 . A pair ( λ f , µ f ) ∈ C 2 is a finite regular eigevalue: rank ( W i ( λ f , µ f )) < nrank ( W i ( λ f , µ f )) , i = 1 , 2 . Geometric multiplicity of ( λ f , µ f ) is 2 � � � rank ( W i ( λ f , µ f )) − nrank ( W i ( λ f , µ f ) . i = 1 The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Our assumptions The two-parameter eigenvalue problem is regular, pencils W 1 ( λ, µ ) and W 2 ( λ, µ ) have full normal rank, nrank ( W i ( λ, µ )) = n i . p 1 ( λ, µ ) = det ( W 1 ( λ, µ )) �≡ 0 , p 2 ( λ, µ ) = det ( W 2 ( λ, µ )) �≡ 0 finitely many eigenvalues ⇒ no common factor of p 1 and p 2 All factors of polynomials p i ( λ, µ ) = det ( W i ( λ, µ )) , i = 1 , 2 depend on both variables λ and µ. ( λ 2 + 1 )( µ + 2 ) ( λ 3 + λ 2 µ + µ 2 + 1 ) The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Eigenvalues of GEP If MEP is nonsingular, eigenvalues coincide with eigenvalues of GEP. Definition A pair ( λ 0 , µ 0 ) ∈ C 2 is a finite regular eigenvalue of matrix pencils ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 if the following is true: a) λ 0 is a finite regular eigenvalue of ∆ 1 − λ ∆ 0 , b) µ 0 is a finite regular eigenvalue of ∆ 2 − µ ∆ 0 , c) there exists a common regular eigenvector z , i.e., z � = 0 such that (∆ 1 − λ 0 ∆ 0 ) z = 0, (∆ 2 − µ 0 ∆ 0 ) z = 0, and z �∈ R (∆ i , ∆ 0 ) for i = 1 , 2. The geometric multiplicity of ( λ 0 , µ 0 ) is dim ( N ) − dim ( N ∩ ( R (∆ 1 , ∆ 0 ) ∪ R (∆ 2 , ∆ 0 ))) , N = ker (∆ 1 − λ 0 ∆ 0 ) ∩ ker (∆ 2 − µ 0 ∆ 0 ) . The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Nonlinear two-parameter eigenvalue problem T 1 ( λ, µ ) x 1 = 0 ( NEP ) T 2 ( λ, µ ) x 2 = 0 , matrix T i ( ., . ) : C × C → C n i × n i is differentiable, i = 1 , 2 . x 1 and x 2 solutions for the eigenvalue ( λ f , µ f ) , x 1 ⊗ x 2 corresponding right eigenvector. left eigenvector y 1 ⊗ y 2 , where y ∗ i T i ( λ f , µ f ) = 0 , i = 1 , 2 . The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Theorem Let ( λ f , µ f ) be an algebraically and geometrically simple eigenvalue of the nonlinear two-parameter eigenvalue problem (NEP) and x 1 ⊗ x 2 , y 1 ⊗ y 2 corresponding right and left eigenvector such that � x 1 � 2 = � x 2 � 2 = � y 1 � 2 = � y 2 � 2 = 1 . Then the matrix � � ∂ T 1 ∂ T 1 y ∗ ∂λ ( λ f , µ f ) x 1 y ∗ ∂µ ( λ f , µ f ) x 1 1 1 M 0 := ∂ T 2 ∂ T 2 y ∗ ∂λ ( λ f , µ f ) x 2 y ∗ ∂µ ( λ f , µ f ) x 2 2 2 is nonsingular. Connection between the Jacobian matrix of the polynomial system and the matrix � � y ∗ y ∗ 1 B 1 x 1 1 C 1 x 1 y ∗ 2 B 2 x 2 y ∗ 2 C 2 x 2 for the nonsingular right definite two-parameter eigenvalue problem (Hochstenbach, Plestenjak, 2003) The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Sketch of the proof � T i ( λ, µ ) � y i define nonsingular S i ( λ, µ ) = x ∗ 0 i define α i ( λ, µ ) = e T n i + 1 S i ( λ, µ ) − 1 e n i + 1 , q i ( λ, µ ) = det ( S i ( λ, µ )) . r i ( λ, µ ) = det ( T i ( λ, µ )) = α i ( λ, µ ) q i ( λ, µ ) , r i ( λ f , µ f ) = 0 and q i ( λ f , µ f ) � = 0 . prove that � � � q 1 ( λ f , µ f ) � ∂ r 1 ∂ r 1 ∂λ ( λ f , µ f ) ∂µ ( λ f , µ f ) = − M 0 . ∂ r 2 ∂ r 2 ∂λ ( λ f , µ f ) ∂µ ( λ f , µ f ) q 2 ( λ f , µ f ) Eigenvalue ( λ f , µ f ) is simple ⇒ Jacobian is nonsingular ⇒ the matrix M 0 is nonsingular The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector ∆ 0 product ( λ ∗ , µ ∗ ) an algebraically simple eigenvalue of MEP x 1 ⊗ x 2 and y 1 ⊗ y 2 corresponding right and left eigenvector � � y ∗ y ∗ 1 B 1 x 1 1 C 1 x 1 � � ( y 1 ⊗ y 2 ) ∗ ∆ 0 ( x 1 ⊗ x 2 ) = � � = 0 � � y ∗ y ∗ 2 B 2 x 2 2 C 2 x 2 � a generalization of the Rayleigh quotient a known result for a right definite MEP, ∆ 0 positive definitive a generalization of Koˇ sir’s result for the nonsingular case The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Characterization of a finite regular eigenvalue Collorary Let λ f be an eigenvalue of the matrix pencil A + λ B with the corresponding right and left eigenvector x and y, respectively. If y ∗ Bx � = 0 then λ f is a finite regular eigenvalue. u ( λ f ) = x (polynomial solution equal to x at λ f ) differentiate ( A + λ B ) u ( λ ) = 0 . Bu ( λ ) + ( A + λ B ) u ′ ( λ ) = 0 ⇒ y ∗ Bu ( λ f ) + y ∗ ( A + λ f B ) u ′ ( λ f ) = y ∗ Bx = 0 � �� � 0 The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector From MEP to coupled GEP Theorem Every algebraically simple eigenvalue ( λ 0 , µ 0 ) of a regular two-parameter eigenvalue problem MEP is a finite regular eigenvalue of the associated pair of generalized eigenvalue problems GEP. an algebraically simple eigenvalue ( λ 0 , µ 0 ) of MEP ( y 1 ⊗ y 2 ) ∗ ∆ 0 x 1 ⊗ x 2 � = 0 a common regular eigenvalue of the associated pair GEP a common regular eigenvalue of GEP → ? an eigenvalue of MEP there is a bidirectonal connection between MEP and GEP The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009

MEP From MEP to a coupled GEP Common regular eigenvector Kronecker chains of A − λ B finite block J p ( α ) ( A − α B ) u 1 = 0 , ( A − α B ) u i + 1 = i = 1 , . . . , p − 1 . Bu i , infinite block N p ( α ) Bu 1 = 0 , i = 1 , . . . , p − 1 . Bu i + 1 = Au i , right singular block L p ( α ) = Au 1 0 , = i = 1 , . . . , p , Au i + 1 Bu i , = 0 Bu p + 1 . left singular block L T p ( α ) = Bu i + 1 , i = 1 , . . . , p − 1 . Au i The Second Najman Conference on Spectral Problems for Operators and Matrices, Dubrovnik, 2009