Eigen-decomposition of Infinite Tridiagonal Matrices: Theory & Numerical Solutions, Slides of Electrical Engineering

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Eigen-decomposition of a class

of

Infinite dimensional tridiagonal matrices

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Outline

 Definition of the problem.

 From finite to infinite dimensions.

 Reduction of the problem to a finite dimensional d.e. and eigen-decomposition problem, using Fourier transform.

 Numerical aspects regarding the solution of the d.e. and the computation of the final (infinite dimensional) eigenvectors.

 Conclusion.

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Eigen-decomposition

Eigenvalues: There is an infinite number.

Eigenvectors: There is an infinite number and each eigenvector

is of infinite size.

Goal: To reduce the infinite dimensional eigen-decomposition

problem into a finite one.

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From finite to infinite dimensions

t t t K t t t

Kr

r

r

Kr

 ^ 

D I A

A A 0

A D I A

Q A D A

A D I A

0 A A

A D I

Q K has dimensions: (2 K +1) N (2 K +1) N,

therefore we have (2 K +1) N eigenvalue-eigenvector pairs.

Typical values: N = 100-1000, K = 5-10.

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t i t i t i

i i (^) i i

l r k l lr k l l r k l

k l k k l k l

 ^ 

A D I A 0

0 A D I A 0

0 A D I A

Consider now the infinite dimensional problem by letting K  

A  i ( k,l+ 1) + ( D + lr I)  i ( k,l ) + A t  i ( k,l- 1) =  i ( k )  i ( k,l ) A  i ( k,l+ 1) + D  i ( k,l ) + A t  i ( k,l- 1) = (  i ( k ) - lr )  i ( k,l ) Docsity.com

Reduction to finite dimensions

A  i ( k,l+ 1)+ D  i ( k,l )+ A t  i ( k,l- 1) = (  i ( k )- lr )  i ( k,l )

A , D : N  N

 i ( k,l ): N  1

i= 1,…, N, k,l= - ,…,

Key Idea

 i ( k ) =  i + kr without loss of generality assume 0   i  r  i ( k,l ) =  i ( l-k )

A  i ( l-k+ 1)+ D  i ( l-k )+ A t  i ( l-k- 1) = (  i -( l-k ) r )  i ( l-k )

A  i ( n+ 1) + ( D-  i I )  i ( n ) + A t  i ( n- 1) = -nr  i ( n ) Docsity.com

Fourier Transform

( ) ( ) jn n

X  x n e^ 

 (^) 



 (^) 

Let …, x (-2), x (-1), x (0), x (1), x (2),… be a real sequence. Then

we define its Fourier Transform as

Important

( ) ( ) jn n

dX nx n e j d

^  

 (^) 



^ 

X (   2 )  X ( )

( ) jn^ jk ( ) n

x n k e^  e ^ X 

 (^) 



^ ^ 

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A  i ( n+ 1)+( D-  i I )  i ( n )+ A t  i ( n- 1) = -rn  i ( n )

i (^ )^ i ( )^ jn n

  n e^ 

 (^) 



  (^) 

( ) j (^) i ( ) ( (^) i ) (^) i ( ) t j (^) i ( ) i d e e jr d

       

A   D  I   A^ ^    

(^1)   ( ) i j^ t^ j i i ( ) d jr e e d

     

 (^)   (^) D  A  A   I 

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Theorem

0

( ) ( ) ( ), (0)

( 2 ) ( ).

dX X X X d

      

 

 

B

B B

Consider the following linear system of d.e.

Let Z ( ) be the transition matrix of the d.e., that is ( ) ( ) ( ), (0)

d d

   

 

Z B Z Z I

then we know that X ( ) = Z ( ) X 0.

Z (2 )  X (^) 0  X 0

The solution X ( ) is periodic if and only if X (2) =X (0)

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(^1 ) (2 )  (^) i (0) e jr^  i^  (2 ) (^) i (0) (^) i (0) ^  Z   Ψ   

(^1 ) (2 )  (^) i (0) ejr^  i^  i (0)

 Ψ   

(^1)   ( ) i j^ t^ j i i ( ) d jr e e d

     

 (^)   (^) D  A  A   I 

(^1)   ( ) j t j ( ), (0) d jr e e d

    

Ψ (^)   (^) D  A  A  Ψ Ψ  I

1 (  ) e jr^   i ( ) ^  Z  Ψ

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Numerical aspects

 Numerical solution of the d.e.

(^1)   ( ) j t j ( ), (0) d jr e e d

    

Ψ (^)   (^) D  A  A  Ψ Ψ  I

 Eigen-decomposition of (2).

 Computation of the Inverse Fourier Transform of  i ( )

where

1 i^ (^ ^ )^ e jr^   i (^ )^ i (0)

^    Ψ 

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Numerical solution of the d.e.

(^1)   ( ) j t j ( ), (0) d jr e e d

    

Ψ (^)   (^) D  A  A  Ψ Ψ  I

( ) ( ) ( ), (0) , ( ) Hermitian

d j d

    

 

Ψ B Ψ Ψ I B

One can show that ( ) is unitary, therefore any numerical solution

should respect this structure. A possible scheme is

1 ( / 2)

( ) ( , ) ( ), (0)

( , )

n n n ej^ ^ ^ 

   

 

  

  

  B

Ψ M Ψ Ψ I

M

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Pade 1, 1 step intgr.

Pade 2, 1 step intgr.

Pade 2, 3 step intgr.

Pade 1, 3 step intgr.

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Eigen-decomposition of  (2  )

Since (2 ) is unitary there are special eigen-decomposition algorithms

that require lower computational complexity than the corresponding

algorithm for the general case.

1 ( ) i^ n ( ) (0)

jr

i n e n i

   

^ 

  Ψ 

From this problem we obtain the pairs  i ,  i (0), i= 1,…, N.

Using the solution ( ) of the differential equation we can compute the

Discrete Fourier Transform of the eigenvectors

Notice that we obtain a sampled version of the required Fourier transform.

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