On singular multiparameter eigenvalue problems Bor Plestenjak - PowerPoint PPT Presentation

On singular multiparameter eigenvalue problems Bor Plestenjak Department of Mathematics University of Ljubljana Joint work with Andrej Muhiˇ c Cortona, September 15-19, 2008 1/23

Outline • Two-parameter eigenvalue problem (2EP) • Singular two-parameter eigenvalue problem • Quadratic two-parameter eigenvalue problem (Q2EP) • An algorithm for the extraction of the common regular part of two matrix pencils • Examples and possible applications Cortona, September 15-19, 2008 2/23

Two-parameter eigenvalue problem • Two-parameter eigenvalue problem: ( A 1 + λB 1 + µC 1 ) x = 0 ( 2EP ) ( A 2 + λB 2 + µC 2 ) y = 0 , where A i , B i , C i are n × n matrices, λ, µ ∈ C , x, y ∈ C n . • Eigenvalue: a pair ( λ, µ ) that satisfies (2EP) for nonzero x and y . • Eigenvector: the tensor product x ⊗ y . • There are n 2 eigenvalues, which are solutions of det ( A 1 + λB 1 + µC 1 ) = 0 det ( A 2 + λB 2 + µC 2 ) = 0 . Cortona, September 15-19, 2008 3/23

Tensor product approach ( A 1 + λB 1 + µC 1 ) x = 0 ( 2EP ) ( A 2 + λB 2 + µC 2 ) y = 0 • On C n ⊗ C n we define n 2 × n 2 matrices ∆ 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 . • 2EP is equivalent to a coupled GEP ∆ 1 z = λ ∆ 0 z ( ∆ ) ∆ 2 z = µ ∆ 0 z, where z = x ⊗ y . • 2EP is nonsingular ⇐ ⇒ ∆ 0 is nonsingular • ∆ − 1 0 ∆ 1 and ∆ − 1 0 ∆ 2 commute. Cortona, September 15-19, 2008 4/23

Numerical methods ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 ( A 1 + λB 1 + µC 1 ) x = 0 ∆ 1 z = λ ∆ 0 z ( 2EP ) ∆ 1 = C 1 ⊗ A 2 − A 1 ⊗ C 2 ( ∆ ) ( A 2 + λB 2 + µC 2 ) y = 0 ∆ 2 z = µ ∆ 0 z ∆ 2 = A 1 ⊗ B 2 − B 1 ⊗ A 2 sir, P. (2005): QZ applied to (∆) . Time complexity: O ( n 6 ) . Hochstenbach, Koˇ Cortona, September 15-19, 2008 5/23

Numerical methods ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 ( A 1 + λB 1 + µC 1 ) x = 0 ∆ 1 z = λ ∆ 0 z ( 2EP ) ∆ 1 = C 1 ⊗ A 2 − A 1 ⊗ C 2 ( ∆ ) ( A 2 + λB 2 + µC 2 ) y = 0 ∆ 2 z = µ ∆ 0 z ∆ 2 = A 1 ⊗ B 2 − B 1 ⊗ A 2 sir, P. (2005): QZ applied to (∆) . Time complexity: O ( n 6 ) . Hochstenbach, Koˇ Algorithms that work directly with matrices A i , B i , C i : • Gradient method: Blum, Curtis, Geltner (1978), Browne, Sleeman (1982) • Newton’s method for eigenvalues: Bohte (1980) • Generalized Rayleigh Quotient Iteration: Ji, Jiang, Lee (1992) • Jacobi-Davidson: – Hochstenbach, P. (2002) for right definite 2EP, – Hochstenbach, Koˇ sir, P. (2005) for nonsingular 2EP, – Hochstenbach, P. (2008) - harmonic extraction Cortona, September 15-19, 2008 5/23

Singular 2EP ∆ 0 = B 1 ⊗ C 2 − C 1 ⊗ B 2 ( A 1 + λB 1 + µC 1 ) x = 0 ∆ 1 z = λ ∆ 0 z ( 2EP ) ∆ 1 = C 1 ⊗ A 2 − A 1 ⊗ C 2 ( ∆ ) ( A 2 + λB 2 + µC 2 ) y = 0 ∆ 2 z = µ ∆ 0 z ∆ 2 = A 1 ⊗ B 2 − B 1 ⊗ A 2 2EP is singular iff ∆ 0 is singular. For singular 2EP, there are no general results linking the eigenvalues of (2EP) and (∆) . We know: ( A 1 + λB 1 + µC 1 ) x = 0 ∆ 1 ( x ⊗ y ) = λ ∆ 0 ( x ⊗ y ) = ⇒ ( A 2 + λB 2 + µC 2 ) y = 0 ∆ 2 ( x ⊗ y ) = µ ∆ 0 ( x ⊗ y ) Cortona, September 15-19, 2008 6/23

Finite regular eigenvalues A pair ( λ, µ ) is a finite regular eigenvalue of (2EP) if: rank ( A i + λB i + µC i ) < max ( s,t ) ∈ C 2 rank ( A i + sB i + tC i ) for i = 1 , 2 . Cortona, September 15-19, 2008 7/23

Finite regular eigenvalues A pair ( λ, µ ) is a finite regular eigenvalue of (2EP) if: rank ( A i + λB i + µC i ) < max ( s,t ) ∈ C 2 rank ( A i + sB i + tC i ) for i = 1 , 2 . A pair ( λ, µ ) is a finite regular eigenvalue of matrix pencils ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 if: 1. rank (∆ 1 − λ ∆ 0 ) < max s ∈ C rank (∆ 1 − s ∆ 0 ) , 2. rank (∆ 2 − µ ∆ 0 ) < max t ∈ C rank (∆ 2 − t ∆ 0 ) , 3. there exists a common eigenvector z in regular parts of ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 such that (∆ 1 − λ ∆ 0 ) z = 0 , (∆ 2 − µ ∆ 0 ) z = 0 . Cortona, September 15-19, 2008 7/23

Finite regular eigenvalues A pair ( λ, µ ) is a finite regular eigenvalue of (2EP) if: rank ( A i + λB i + µC i ) < max ( s,t ) ∈ C 2 rank ( A i + sB i + tC i ) for i = 1 , 2 . A pair ( λ, µ ) is a finite regular eigenvalue of matrix pencils ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 if: 1. rank (∆ 1 − λ ∆ 0 ) < max s ∈ C rank (∆ 1 − s ∆ 0 ) , 2. rank (∆ 2 − µ ∆ 0 ) < max t ∈ C rank (∆ 2 − t ∆ 0 ) , 3. there exists a common eigenvector z in regular parts of ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 such that (∆ 1 − λ ∆ 0 ) z = 0 , (∆ 2 − µ ∆ 0 ) z = 0 . Conjecture. Finite regular eigenvalues of (2EP) = finite regular eigenvalues of (∆) . Cortona, September 15-19, 2008 7/23

Quadratic 2EP ( A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 ) x = 0 ( Q2EP ) ( A 2 + λB 2 + µC 2 + λ 2 D 2 + λµE 2 + µ 2 F 2 ) y = 0 , where A i , B i , . . . , F i are n × n matrices, ( λ, µ ) is an eigenvalue, and x ⊗ y is the corresponding eigenvector. In the generic case the problem has 4 n 2 eigenvalues that are solutions of det ( A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 ) = 0 det ( A 2 + λB 2 + µC 2 + λ 2 D 2 + λµE 2 + µ 2 F 2 ) = 0 . Jahrlebring (2008): Q2EP of a simpler form, with some of the terms λ 2 , λµ, µ 2 missing, appears in the study of linear time-delay systems for the single delay. Cortona, September 15-19, 2008 8/23

Linearization ( A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 ) x = 0 ( Q2EP ) ( A 2 + λB 2 + µC 2 + λ 2 D 2 + λµE 2 + µ 2 F 2 ) y = 0 Cortona, September 15-19, 2008 9/23

Linearization ( A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 ) x = 0 ( Q2EP ) ( A 2 + λB 2 + µC 2 + λ 2 D 2 + λµE 2 + µ 2 F 2 ) y = 0 Vinnikov (1989): It follows from the theory on determinantal representations that one could write Q2EP as a two-parameter eigenvalue problem with 2 n × 2 n matrices. Since there is no construction this is just a theoretical result. Cortona, September 15-19, 2008 9/23

Linearization ( A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 ) x = 0 ( Q2EP ) ( A 2 + λB 2 + µC 2 + λ 2 D 2 + λµE 2 + µ 2 F 2 ) y = 0 Vinnikov (1989): It follows from the theory on determinantal representations that one could write Q2EP as a two-parameter eigenvalue problem with 2 n × 2 n matrices. Since there is no construction this is just a theoretical result. Best we can do is to write Q2EP as a two-parameter eigenvalue problem with 3 n × 3 n matrices: ✵ ✷ ✸ ✷ ✸ ✷ ✸ ✶ ✷ ✸ 0 0 0 A 1 B 1 C 1 D 1 E 1 F 1 x ✺ + λ ✺ + µ ❅ ✹ ✹ ✹ ✺ ❆ ✹ ✺ 0 − I 0 I 0 0 0 0 0 λx = 0 0 0 − I 0 0 0 I 0 0 µx ✵ ✷ ✸ ✷ ✸ ✷ ✸ ✶ ✷ ✸ 0 0 0 A 2 B 2 C 2 D 2 E 2 F 2 y ❅ ✹ ✺ + λ ✹ ✺ + µ ✹ ✺ ❆ ✹ ✺ 0 0 0 0 0 0 0 = 0 . − I I λy 0 0 0 0 0 0 0 − I I µy Cortona, September 15-19, 2008 9/23

Weak linearization If we multiply the matrix of the first equation ✷ ✸ B 1 + λD 1 C 1 + λE 1 + µF 1 A 1 ✹ ✺ λI − I 0 µI 0 − I from left by the unimodular polynomial ✷ ✸ ✷ ✸ B 1 + λD 1 0 0 C 1 + λE 1 + µF 1 I I ✹ ✺ ✹ ✺ E ( λ, µ ) = 0 0 0 0 I I 0 0 0 0 I I and from right by the unimodular polynomial ✷ ✸ ✷ ✸ 0 0 0 0 I I ✹ ✺ ✹ ✺ F ( λ, µ ) = 0 I 0 λI I 0 µI 0 I 0 0 I we obtain ✷ ✸ A 1 + λB 1 + µC 1 + λ 2 D 1 + λµE 1 + µ 2 F 1 0 0 ✹ ✺ . 0 0 I 0 0 I Cortona, September 15-19, 2008 10/23

Linearization is a singular 2EP ✷ ✸ ✷ ✸ ✷ ✸ 0 0 0 A 1 B 1 C 1 D 1 E 1 F 1 A (1) + λB (1) + µC (1) ✹ ✺ + λ ✹ ✺ + µ ✹ ✺ = 0 0 0 0 0 0 0 − I I 0 0 0 0 0 0 0 − I I ✷ ✸ ✷ ✸ ✷ ✸ A 2 B 2 C 2 0 D 2 E 2 0 0 F 2 A (2) + λB (2) + µC (2) ✺ + λ ✺ + µ ✹ ✹ ✹ ✺ . = 0 0 0 0 0 0 0 − I I 0 0 0 0 0 0 0 − I I The matrices of the corresponding pair of generalized eigenvalue problems are B (1) ⊗ C (2) − C (1) ⊗ B (2) , ∆ 0 = C (1) ⊗ A (2) − A (1) ⊗ C (2) , ∆ 1 = A (1) ⊗ B (2) − B (1) ⊗ A (2) . ∆ 2 = Lemma. In the generic case (matrices D 1 , D 2 , F 1 , F 2 are all nonsingular) it follows: 1. rank (∆ 1 ) = rank (∆ 2 ) = 8 n 2 , 2. rank (∆ 0 ) = 6 n 2 , 3. det ( α 0 ∆ 0 + α 1 ∆ 1 + α 2 ∆ 2 ) = 0 for all α 0 , α 1 , α 2 . Cortona, September 15-19, 2008 11/23

Regular eigenvalues In the generic case (matrices D 1 , D 2 , F 1 , F 2 are all nonsingular), we have: ✷ ✸ ✷ ✸ 0 0 ✹ ✺ ⊗ ✹ ✺ , 1. A basis for ker (∆ 1 ) is e i e j i, j = 1 , . . . , n. 0 0 ✷ ✸ ✷ ✸ 0 0 D − 1 D − 1 ✹ ✺ ⊗ ✹ ✺ , 2. A basis for ker (∆ 2 ) is i, j = 1 , . . . , n. 1 E 1 e i 2 E 2 e j − e i − e j 3. ker (∆ i ) ⊂ ker (∆ 0 ) for i = 1 , 2 . A basis for the remaining vectors in ker (∆ 0 ) is ✷ ✸ ✷ ✸ 0 0 ✹ D − 1 ✺ ⊗ ✹ D − 1 ✺ , 1 ( E 1 − F 1 ) e i 2 ( E 2 − F 2 ) e j i, j = 1 , . . . , n. − e i − e j The eigenvalues of Q2EP are regular eigenvalues of the coupled matrix pencils Theorem. ∆ 1 − λ ∆ 0 and ∆ 2 − µ ∆ 0 from the weak linearization. Cortona, September 15-19, 2008 12/23