Classifications of quasiseparable matrices in terms of recurrence - PowerPoint PPT Presentation

Tom Bella Classifications of quasiseparable matrices in terms of recurrence relations Tom Bella Department of Mathematics University of Rhode Island Joint work with Yuli Eidelman, Israel Gohberg, Vadim Olshevsky, & Pavel Zhlobich Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 1

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Moment Matrices ➠ Hankel matrices. Defined by O ( n ) parameters { h k } .   h 0 h 2 · · · h n − 1 h 1 .  ...  . h 1 h 2   .   � � ...   H = = h k + j h 2 h 2 n − 3     . ... .   h 2 n − 3 h 2n − 2 .   · · · h n − 1 h 2 n − 3 h 2n − 2 h 2 n − 1 ➠ Toeplitz matrices. Defined by O ( n ) parameters { t k } .   t 0 · · · · · · t − n +1 t − 1   . .   t 1 t 0 t − 1 .   � �   . . ... ... ... C = = . . t k − j   . .     . ... .   t 0 t − 1 .   · · · · · · t n − 1 t 1 t 0 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 2

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Moment Matrices ➠ Both of these classes of matrices are related to orthogonal polynomials . ➠ For a given inner product, the moment matrix is   � 1 , x 2 � � 1 , x n � � 1 , 1 � � 1 , x � . . .   � 1 , x 2 � � x, x n � � x, 1 � � x, x � . . .     � x 2 , 1 � � x 2 , x � � x 2 , x 2 � � x 2 , x n � M = [ � x k , x j � ] = . . .     . . . .   . . . . . . . .   � x n , x 2 � � x n , 1 � � x n , x � � x n , x n � . . . ➠ For an inner product defined by integration on the real line, � b � b p ( x ) q ( x ) w 2 ( x ) dx, x ( k + j ) w 2 ( x ) dx, � x k , x j � = � p ( x ) , q ( x ) � = ⇒ a a and M is Hankel. ➠ Hankel matrices are related to real–orthogonal polynomials. Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 3

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Moment Matrices ➠ Both of these classes of matrices are related to orthogonal polynomials . ➠ For a given inner product, the moment matrix is   � 1 , x 2 � � 1 , x n � � 1 , 1 � � 1 , x � . . .   � 1 , x 2 � � x, x n � � x, 1 � � x, x � . . .     � x 2 , 1 � � x 2 , x � � x 2 , x 2 � � x 2 , x n � M = [ � x k , x j � ] = . . .     . . . .   . . . . . . . .   � x n , x 2 � � x n , 1 � � x n , x � � x n , x n � . . . ➠ For an inner product defined by integration on the unit circle, � π � π p ( e iθ ) · q ( e iθ ) w 2 ( θ ) dθ ⇒ � x k , x j � = x ( k − j ) w 2 ( θ ) dθ, � p ( x ) , q ( x ) � = − π − π and M is Toeplitz. ➠ Toeplitz matrices are related to Szeg¨ o polynomials. Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 3

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Recurrent Matrices ➠ Tridiagonal matrices. Defined by O ( n ) parameters.   δ 1 γ 2 0 · · · 0   . ... .   γ 2 δ 2 γ 3 .     ... T =   0 0 γ 3 δ 3     . ... ... ... .   γ n .   0 · · · 0 γ n δ n ➠ Unitary Hessenberg matrices. Defined by O ( n ) parameters.   − ρ 1 ρ 0 ∗ − ρ 2 µ 1 ρ 0 ∗ · · · − ρ n µ n − 1 ...µ 1 ρ 0 ∗   − ρ 2 ρ 1 ∗ · · · − ρ n µ n − 1 ...µ 2 ρ 1 ∗ µ 1     . . ... . .   U = . .     . ... .   − ρ n µ n − 1 ρ n − 2 ∗ .   0 · · · − ρ n ρ n − 1 ∗ µ n − 1 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 4

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Recurrent Matrices ➠ Both of these classes of matrices are related to orthogonal polynomials . ➠ The system of polynomials defined by r k ( x ) = det( xI − T ) ( k × k ) where   0 · · · 0 δ 1 γ 2 . ...   . γ 2 δ 2 γ 3 .     ...   T = 0 0 γ 3 δ 3     . ... ... ...  .  γ n .   0 · · · 0 γ n δ n are real–orthogonal polynomials. Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 5

Introduction Tom Bella Orthogonal Polynomials Related to Structured Matrices Recurrent Matrices ➠ Both of these classes of matrices are related to orthogonal polynomials . ➠ The system of polynomials defined by r k ( x ) = det( xI − U ) ( k × k ) where   − ρ 1 ρ 0 ∗ − ρ 2 µ 1 ρ 0 ∗ · · · − ρ n µ n − 1 ...µ 1 ρ 0 ∗   − ρ 2 ρ 1 ∗ · · · − ρ n µ n − 1 ...µ 2 ρ 1 ∗ µ 1     . . ...  . .  U = . .     . ... .   − ρ n µ n − 1 ρ n − 2 ∗ .   0 · · · − ρ n ρ n − 1 ∗ µ n − 1 are Szeg¨ o polynomials. Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 5

Introduction Tom Bella Generalizations of these Structures Matrix class Generalized class Hankel matrices matrices with displacement structure Toeplitz matrices tridiagonal matrices ????????????????????????????? unitary Hessenberg matrices Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 6

Introduction Tom Bella Generalizations of these Structures Matrix class Generalized class Hankel matrices matrices with displacement structure Toeplitz matrices tridiagonal matrices quasiseparable matrices unitary Hessenberg matrices Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 6

Introduction Tom Bella Quasiseparable Matrices ➠ Definition. A matrix C is ( H, m ) –quasiseparable if it is strongly upper Hessenberg (nonzero subdiagonals, zeros below that) and max Rank C 12 = m where the maxima are taken over all symmetric partitions of the form � � ∗ C 12 C = ∗ ➠ Previous work. Chandrasekaran, Eidelman, Fasino, Gemignani, Gohberg, Gu, Kailath, Koltracht, Mastronardi, Olshevsky, Van Barel, Vandebril... ➠ A system of polynomials related to an ( H, m ) –quasiseparable matrix C as character- istic polynomials of principal submatrices of C , i.e. r k ( x ) = det( xI − C k × k ) will be called ( H, m ) –quasiseparable polynomials. Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 7

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Tridiagonal   0 0 0 d 1 g 1   0 0 q 1 d 2 g 2     C = 0 0 q 2 d 3 g 3       0 0 q 3 d 4 g 4   0 0 0 q 4 d 5 ➠ The system of polynomials r k ( x ) = det( xI − C k × k ) associated with C are real orthogonal polynomials with recurrence relations r k ( x ) = 1 ( x − d k ) r k − 1 ( x ) − g k − 1 r k − 2 ( x ) q k q k ➠ The matrix C is ( H, 1) –quasiseparable . Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 8

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Tridiagonal   0 0 0 d 1 g 1   0 0 q 1 d 2 g 2     C = 0 0 q 2 d 3 g 3       0 0 q 3 d 4 g 4   0 0 0 q 4 d 5 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 8

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Tridiagonal   0 0 0 d 1 g 1   0 0 q 1 d 2 g 2     C = 0 0 q 2 d 3 g 3       0 0 q 3 d 4 g 4   0 0 0 q 4 d 5 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 8

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Tridiagonal   0 0 0 d 1 g 1   0 0 q 1 d 2 g 2     C = 0 0 q 2 d 3 g 3       0 0 q 3 d 4 g 4   0 0 0 q 4 d 5 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 8

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Tridiagonal   0 0 0 d 1 g 1   0 0 q 1 d 2 g 2     C = 0 0 q 2 d 3 g 3       0 0 q 3 d 4 g 4   0 0 0 q 4 d 5 Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 8

Introduction Tom Bella Important Special Cases of Quasiseparable Matrices Unitary Hessenberg   − ρ ∗ − ρ ∗ − ρ ∗ − ρ ∗ − ρ ∗ 0 ρ 1 0 µ 1 ρ 2 0 µ 1 µ 2 ρ 3 0 µ 1 µ 2 µ 3 ρ 4 0 µ 1 µ 2 µ 3 µ 4 ρ 5   − ρ ∗ − ρ ∗ − ρ ∗ − ρ ∗ µ 1 1 ρ 2 1 µ 2 ρ 3 1 µ 2 µ 3 ρ 4 1 µ 2 µ 3 µ 4 ρ 5     C = 0 − ρ ∗ − ρ ∗ − ρ ∗ µ 2 2 ρ 3 2 µ 3 ρ 4 2 µ 3 µ 4 ρ 5       0 0 − ρ ∗ − ρ ∗ µ 3 3 ρ 4 3 µ 4 ρ 5   0 0 0 − ρ ∗ µ 4 4 ρ 5 ➠ The system of polynomials r k ( x ) = det( xI − C k × k ) associated with C are the Szeg¨ o polynomials with recurrence relations � � � � � � G k ( x ) = 1 1 − ρ ∗ G k − 1 ( x ) k µ k r k ( x ) − ρ k 1 xr k − 1 ( x ) ➠ The matrix C is ( H, 1) –quasiseparable . Structured Linear Algebra Problems, Cortona, Italy, 2008 Page 9