-pseudo involutions

Gi-Sang Cheon, Sung-Tae Jin and Hana KimSungkyunkwan University

2009.08.21

m

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

m

( ) (1, )m z

m

m

Riordan group

Riordan group (L. Shapiro, 1991)

with

A Riordan matrix is an infinite lower triangular matrix whose th columnhas the GF for .

0 {0,1,2, }, ( ), ( ) [[ ]]g z f z z N C

(0) 0, (0) 0, (0) 0g f f

( ( ), ( ))g z f z

k ( ) ( ) kg z f z 0kN

= the set of all Riordan matrices

forms a group under the Riordanmultiplication defined by

is called the Riordan group.

*( ( ), ( ))*( ( ), ( )) ( ( ) ( ( )), ( ( ))).g z f z h z l z g z h f z l f z

( ,*)

An involution & pseudo involution

A matrix is called an involution if i.e.,

An involution in the Riordan group is called a Riordan involution.

A 2 ,A I1 .A A

If a Riordan matrix satisfies i.e., then is called a pseudo involution.

0, ,[ ]n k n kR r N

2( ) ,RM I

1 0 0 00 1 0 0

(1, ) 0 0 1 00 0 0 1

M z

1,[( 1) ]n kn kR MRM r

R

The centralizer of

The centralizer of in the Riordan group is

which is the checkerboard subgroup.

(1, )M z

( )m

( ) | ( ) [[ ]], (0) 1 ( 1)mzf z f z z f m C

( ) {( ( ), ( )) | ( ), ( ) }C M g z f z zg z f z

For let

where is a root of i.e.,

1,2,3, ,m

2

3

1 0 0 00 0 0

: (1, ) 0 0 00 0 0

z

C(2 1)exp

(2 1) (2 1)cos sin ( 0,1, , 1).

k imk ki k mm m

1,mz

( )m

Theorem 1. The centralizer of in the Riordan group is

Note :• • is a subgroup of

(1)( ) ( )C C M

( )m

( )( ).mC ( 1)( )mC

( )( ) {( ( ), ( )) | ( ), ( ) } ( ).mC g z f z zg z f z m N

- pseudo involutions

For each we say that a Riordan matrix is a - pseudo involution if where

if (mod )

otherwise.

m

mN

0, ,( ( ), ( )) [ ]n k n kQ g z f z q N m1

,[ ]n kQ q

,,

,

( 1)n km

n kn k

n k

qqq

mn k

Example (2-pseudo involution)

where2 2 41 1 6( )

2z z zs z

z

10 12 0 10 4 0 16 0 6 0 10 16 0 8 0 122 0 30 0 10 0 1

S

1

( ) , ( )s zS s zz

Theorem 2. If is a - pseudo involutionthen .

Corollary 3. If is a - pseudo involution then

( ( ), ( ))Q g z f z m

Q m

( ), ( )zg z f z

1 ( ) ( ) 1( ( ), ( )).m mQ Q g z f z

Corollary 4. If is a - pseudo involution then

Corollary 5. If is a - pseudo involution then the orderof is in the Riordan group.Q

( )mQ

m

2m

2mQ ( )( ).mQ C

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 ).

m

mN

1 2, ,m m N

0k N 1 2 2km m 12k

m

Theorem 6.The relation is an equivalence relation on

For each it is sufficient to consider - pseudo involutions in the Riordan group.

0 ,kN2k

.O

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

0, ,[ ]n k n kR r N

0 1 2( , , , )A a a a

m

0 1 2( , , , )Z z z z 0 00, 0a z

1, 1 , 00(i) ( , ),n k j n k jjr a r k n

N

1,0 , 00(ii) ( )n j n jj

r z r n

N

A -sequence of R

-sequence of Z R

Theorem 7. is a - pseudo involution with the -seq.GF iff has the -seq. GF where is a root of

AQ

A1Q( )A z

m

1.mz

( )A z

A Riordan matrix has a -sequence if

,( ( ), ( )) [ ]n kR g z f z r

m

1m

1, 1 , 00

( , ).n k j n mj k mjj

r s r n k

N0 1 2( , , , )s s s

2m

( )( ( ))m m f zz f zz

m ( )z-seq. GF =

1n

1k

Theorem 8. Let be a Riordan matrixsatisfying where is a root of iff there exists a -sequence GF suchthat

( ) ( ( )) 1g z g f z

1( ) .( )

zz

m1.mz

( ( ), ( ))R g z f z

2mRThen is a - pseudo involution( )z

Example (4-pseudo involution) Consider the - sequence GF

(the GF for twice Fibonacci numbers)

Let where satisfies( )f z( ) , ( )f zB f zz

2 2 ( )( ( )) .f zz f zz

22 3 4

2

1( ) 1 2 2 4 6 .1z zz z z z zz z

2

Then we have

1

10 10 0 10 0 0 12 0 0 0 10 4 0 0 0 10 0 6 0 0 0 1

.0 0 0 8 0 0 0 1

10 0 0 0 10 0 0 0 10 24 0 0 0 12 0 0 0 10 0 42 0 0 0 14 0 0 0 10 0 0 64 0 0 0 16 0 0 0 1

66 0 0 0 90 0 0 0 18 0 0 0 1

B

Theorem 9. If is a - pseudo involution then is alsoa - pseudo involution for

Theorem 10. If is a - pseudo involution then isa - pseudo involution for any diagonalmatrix

mQ

.nZm

nQ

Q m

m

1DQD

(1, ), \{0}.D az a C

Application to Toeplitz systems

We define a - pseudo involution of the general linear group by such that for where is the principal submatrix of

A

1 ( ) ( )m mn nA A

AGL( , )n C

m

1m( ).m

( )mn

n n

Let us consider the problem where

is a Toeplitz matrix.

A x b

0 1 2 3

1 0 1 2

2 1 0 1

2 1 03

a a a aa a a a

A a a a aa a a a

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

A

A x b

The commutator of plays an important role to get - pseudo involutions.

Theorem 11. Let Then is a pseudoinvolution.

(1)[ , ]nA GL( , ).A n C

m

, GL( , )A B n C

1 1[ , ]A B ABA B

Example Let

1 2 0 3 3 2 3 1 5 13 3 2 2 1 0 1 4 2 22 0 1 4 1 2 4 6 0 3

0 1 0 3 1 1 2 3 4 05 3 2 2 0 1 2 0 1 6

GL(10, ).3 1 5 3 2 1 3 6 7 11 2 1 1 3 2 7 2 1 06 3 0 1 5 1 2 0 3 50 4 2 4 1 5 2 1 4 13 1 7 2 0 0 1 2 0 1

A

C

Then

522259 70416 203624 693110 71292 46486 98326 66892 185156 23188202931 202931 202931 202931 202931 202931 202931 202931 202931 2029311445218 6275473 6173881 457797 2534796 776431202931 405862 405862 202931 202931 202931

1476414 142896 933605 567437202931 202931 202931 405862

278746 658156 272501 4700 464796 87672 35344 36806 218530 165318202931 202931 202931 202931 202931 202931 202931 202931 202931 202931

2615980 4958353 590547202931 405862

9 1480160 1831524 649875 1146598 332480 1210577 818999405862 202931 202931 202931 202931 202931 202931 405862

4844858 15028195 15275193 640591 5835889 1824723 3453548 569202931 405862 405862 202931 202931 202931 202931

102 2607977 1771137

202931 202931 4058621652196 722923 907657 1446804 402240 229387 13370 64518 651868 458915202931 202931 202931 202931 202931 202931 202931 202931 202931 202931385820 1102641 1278087202931 405862 4058

49061 393168 135413 467137 91202 81939 72571

62 202931 202931 202931 202931 202931 202931 4058621837816 426716 1528884 1867974 338928 532380 231622 303353 817870 3202931 202931 202931 202931 202931 202931 202931 202931 202931

87530

2029315674048 12768749 14536075 2238489 4781508 1728079 2855550 764868 2620326 1873523202931 405862 405862 202931 202931 202931 202931 202931 202931 405862

304616 680421 396131 723903 39202931 405862 405862 202931

3360 81569 189614 59682 24107 444247

202931 202931 202931 202931 202931 405862

(1)10[ , ]A

Then

1044518 140832 407248 1386220 142584 92972 196652 133784 370312 463762890436 6275473 6173881 915594 5069592 1552862 2952828 285792 1867210 567437557492 1316312 545002 9400 929592 175344 70688 73612 437060 3

1405862

306365231960 4958353 5905479 2960320 3663048 1299750 2293196 664960 2421154 8189999689716 15028195 15275193 1281182 11671778 3649446 6907096 1138204 5215954 17711373304392 1445846 1815314 2893608 804480 45877

4 26740 129036 1303736 917830

771640 1102641 1278087 98122 786336 270826 934274 182404 163878 725713675632 853432 3057768 3735948 677856 1064760 463244 606706 1635740 77506011348096 12768749 14536075 4476978 9563

016 3456158 5711100 1529736 5240652 1873523

609232 680421 396131 1447806 786720 163138 379228 119364 48214 444247

(1)10[ , ]A 1

The centralizer of in :

Theorem 12. Let Then if and onlyif is a - pseudo involution.

( ) ( ) ( ) ( )( ) GL( , ) | [ , ]m m m mn n n n nC A n A A A I C

GL( , )n C( )mn

0, ,

,

[ ] GL( , ) |

0 if (mod 2 )

i j i j

i j

A a n

a i j m

N C

(2 )[ , ]mnA 2m

( )( )mnA C GL( , ).A n C

Theorem 13. Let If is a - pseudoinvolution of Toeplitz type then (mod ).0n

m

2m

( )[ , ]mnA GL( , ).A n C

Theorem 14. For and let

Then is a pseudo involution of Toeplitztype.

1n

1 0 01 0

0 1 0GL(2 , ).

0 10 0 1

a aa a aa a

A na a aa a

C

(1)2[ , ]nA

1 ,a n

In fact,(1)2

2 2

2 2 2

2 2

2 2 2 2

2 2

[ , ]

1 2 2 2 22 1 2 2 2

2 2 1 2 21 .1 ( ) 2 2 2 1 2

2 2 1

nA

na a na a aa na a na nana a na a a

na a na a na na

a na na

Theorem 15. Let be a Toeplitz matrix. Then is a pseudo involution if and only if the Kronecker product is a - pseudoinvolution for

GL(2 , )A n C

A

mmA I

1.m

Example Let us consider

A

1 01 0

0 10 1

a aa aa a

a a

1 11 04 41 11 04 4 .

1 10 14 4

1 10 14 4

Then

is a pseudo involution.

1(1)4

4 2 1 22 4 2 11[ , ] 3 1 2 4 22 1 2 4

A

:B

is a 3 - pseudo involution.

1( )3

4 0 0 2 0 0 1 0 0 2 0 00 4 0 0 2 0 0 1 0 0 2 00 0 4 0 0 2 0 0 1 0 0 22 0 0 4 0 0 2 0 0 1 0 00 2 0 0 4 0 0 2 0 0 1 00 0 2 0 0 4 0 0 2 0 0 111 0 0 2 0 0 4 0 0 2 0 030 1 0 0 2 0 0 4 0 0 2 00 0 1 0 0 2 0 0 4 0 0 22 0 0 1 0 0 2 0 0 4 0 00 2 0 0 1 0 0 2 0 0 4 00 0 2 0 0 1 0 0 2 0 0 4

B I

Thank you for your Thank you for your attention.attention.