-pseudo involutions Gi-Sang Cheon, Sung-Tae Jin and Hana Kim Sungkyunkwan University 2009.08.21
Contents • Riordan group - An involution and pseudo involution - The centralizer of • -pseudo involutions - Classification of -pseudo involutions - Characterization of -pseudo involutions • Application to Toeplitz systems
Riordan group Riordan group (L. Shapiro, 1991) • with • A Riordan matrix is an infinite lower triangular matrix whose th column has the GF for .
= the set of all Riordan matrices • forms a group under the Riordan multiplication defined by • is called the Riordan group.
An involution & pseudo involution • A matrix is called an involution if i.e., • An involution in the Riordan group is called a Riordan involution.
If a Riordan matrix satisfies i.e., then is called a pseudo involution.
The centralizer of • The centralizer of in the Riordan group is which is the checkerboard subgroup.
For let where is a root of i.e.,
Theorem 1. The centralizer of in the Riordan group is • Note : • is a subgroup of
- pseudo involutions • For each we say that a Riordan matrix is a - pseudo involution if where if (mod ) otherwise.
Theorem 2. If is a - pseudo involution then . • Corollary 3. If is a - pseudo involution then
Corollary 4. If is a - pseudo involution then • Corollary 5. If is a - pseudo involution then the order of is in the Riordan group.
Classification of - pseudo involutions • = the set of all - pseudo involutions for each • = the collection of ’s • Define a relation on by for iff such that (mod ).
Theorem 6. The relation is an equivalence relation on • For each it is sufficient to consider - pseudo involutions in the Riordan group.
-sequence of -sequence of Characterization of - pseudo involutions • Theorem (Rogers ‘78, Sprugnoli ‘94) An i.l.t.m. is a Riordan matrix iff two sequences and with such that
Theorem 7. is a - pseudo involution with the -seq. GF iff has the -seq. GF where is a root of
-seq. GF = • A Riordan matrix has a -sequence if
Then is a - pseudo involution • Theorem 8. Let be a Riordan matrix satisfying where is a root of iff there exists a -sequence GF such that
Example (4-pseudo involution) Consider the - sequence GF (the GF for twice Fibonacci numbers) Let where satisfies
Theorem 9. If is a - pseudo involution then is also a - pseudo involution for • Theorem 10. If is a - pseudo involution then is a - pseudo involution for any diagonal matrix
Application to Toeplitz systems • We define a - pseudo involution of the general linear group by such that for where is the principal submatrix of
Let us consider the problem where is a Toeplitz matrix.
When is symmetric and positive definite Toeplitz matrix, there are three algorithms to solve the system : • Durbin’s algorithm • Levinson’s algorithm • Trench’s algorithm
The commutator of plays an important role to get - pseudo involutions. • Theorem 11. Let Then is a pseudo involution.
The centralizer of in : • Theorem 12. Let Then if and only if is a - pseudo involution.
Theorem 13. Let If is a - pseudo involution of Toeplitz type then (mod ).
Theorem 14. For and let Then is a pseudo involution of Toeplitz type.
Theorem 15. Let be a Toeplitz matrix. Then is a pseudo involution if and only if the Kronecker product is a - pseudo involution for
Example Let us consider
Then is a pseudo involution.