The Two-Sided Eigenvalue Problem

The two-sided eigenvalue problem is the problem of finding non-trivial solutions to the system of equations

Aw=λBw (1.6.22)

(the trivial solution being w = 0) where A is an n×n symmetric matrix and B ann×n positive definite symmetric matrix. A non-zero vector, w, satisfying equa-

tion (1.6.22) for a given constant,λ, is called an eigenvector of the two-sided eigen- value problem associated with the eigenvalue,λ, of the two-sided eigenvalue problem. It is clear that when B =I, the two-sided eigenvalue problem reduces to the usual known (one-sided) eigenvalue problem, that is the problem of finding non-trivial solutions to the system of equations in (1.6.22).

In order for a solutionwof equation (1.6.22) to be a non-zero vector, λmust be a root of the characteristic equation

∣A−λB∣=0.

This characteristic equation is a polynomial of degree n and therefore has n roots (not necessarily all distinct), each of which is associated with a non-trivial solution of equation (1.6.22). Note that, similar to the one-sided eigenvalue problem, the eigenvectors of the two-sided eigenvalue problem are only uniquely defined up to a scalar multiple, that is, if w is an eigenvector of the two-sided eigenvalue problem associated with the eigenvalue,λ, then any scalar multiple ofwis also an eigenvector of the two-sided eigenvalue problem associated with the eigenvalue,λ. This is shown below:

Aw=λBw

Ð→A(cw)=λB(cw) .

In the remainder of this thesis, it will be assumed that every eigenvector,w, of the two-sided eigenvalue problem is normalised such that

w′Bw=1.

The eigenvectors of the two-sided eigenvalue problem in (1.6.22) that satisfy this constraint are uniquely defined up to multiplication by minus one.

It will now be shown that whenλi ≠λj, the solutionsw(i)andw(j)ofAw=λiBw

andAw=λjBwrespectively are orthogonal in the metric, B, that isw₍′_j)Bw(i)=0.

Let w₍i) and w(j) be solutions of Aw = λiBw and Aw = λjBw respectively with

λi≠λj. This implies that the following two equations hold:

w₍′_i)Aw₍j)=λjw′_(i)Bw(j)

Sincew′_(i)Aw₍j)=w′₍j)Aw(i), it follows that

λjw′_(i)Bw(j)=λiw′_(j)Bw(i)

and since λi≠λj, it follows that

w₍′_i)Bw₍j)=w(′j)Bw(i)=0.

The set ofnsolutions of the system of equations in (1.6.22) can be simultaneously represented by

AW=BWΛ (1.6.23)

whereΛis ann×ndiagonal matrix, the diagonal elements of which are the eigenval- ues of the two-sided eigenvalue problem andW is ann×n matrix withjth column vector, w₍j), equal to the eigenvector of the two-sided eigenvalue problem which is

associated with the eigenvalue given by the jth diagonal element of Λ. Note that the diagonal elements ofΛ can be ordered in any way, as long as the column vectors of W are ordered accordingly. In the remainder of this thesis it will be assumed that the eigenvalues of a two-sided eigenvalue problem in (1.6.22) are ordered in descending order as the diagonal elements of Λ. For convenience the jth column vector ofW, that is the eigenvector of the two-sided eigenvalue problem associated with thejth largest eigenvalue, λj, will be referred to as the jth eigenvector of the

two-sided eigenvalue problem. Note that since

w′_(i)Bw₍j)={ 1₀ if_otherwisei=j

it follows that

W′BW=I.

is obtained:

W′AW=Λ

is obtained. It is evident that given any symmetric matrixA and positive definite matrixB, there exists a non-singular transformation matrix, W, which is such that W′BW=IandW′AW=ΛwhereΛis a diagonal matrix with its diagonal elements written in non-increasing order. The matrixW is formed from the solutions to the two-sided eigenvalue problem - thejth column vector of W, w₍j), is the solution of

Aw=λjBw

whereλj is thejth largest eigenvalue of the two-sided eigenvalue problem in (1.6.22).

The vector w₍j) is called the eigenvector of the two-sided eigenvalue problem in

(1.6.22) corresponding to the jth largest eigenvalue of that two-sided eigenvalue problem.

The only task left is to find the solutions of the two-sided eigenvalue problem. Note that sinceB is positive definite, it has an unique square root matrix. Let the svd ofB be given by

B=VΛV′.

It follows that the square root matrix ofB is given by

B1/2=VΛ1/2V where [Λ1/2] ij ={ [ Λ]1_ii/2 if i=j 0 otherwise

(see Section 1.6.2). By manipulating the expression AW = BWΛ algebraically using B =B1/2_B1/2 _{and B}1/2_B−1/2 _{=I, it can be shown that the ith column vector}

ofB−1/2_{U, where U is the matrix theith column vector of which is the eigenvector}

the two-sided eigenvalue problem: AW=BWΛ (1.6.24) Ð→AW=B1/2B1/2WΛ Ð→B−1/2AW=B1/2WΛ Ð→B−1/2AB−1/2B1/2W=B1/2WΛ Ð→(B−1/2AB−1/2)(B1/2W)=(B1/2W)Λ. (1.6.25)

It it evident that the column vectors of B1/2_{Ware the eigenvectors of B}−1/2_AB−1/2

with the corresponding eigenvalues given by the diagonal elements of Λ. Let U = B1/2_{W so that:}

W=B−1/2U.

This shows that n solutions of Aw = λBw can be found by firstly finding n ei- genvectors of B−1/2_AB−1/2 _{and then pre-multiplying the matrix with jth column}

vector given by the eigenvector of B−1/2_AB−1/2 _{that corresponds to the jth largest}

eigenvalue of B−1/2_AB−1/2_{, by B}−1/2_{. It is evident from equation (1.6.25) that the}

eigenvalues of the two-sided eigenvalue problem in (1.6.22) are identical to the ei- genvalues of the matrix B−1/2_AB−1/2_{. This means that the number of non-zero}

eigenvalues of the two-sided eigenvalue problem in equation (1.6.24) is equal to the rank of the matrix B−1/2_AB−1/2_{, which is equal to the rank of the matrix A since}

B−1/2 _{is a non-singular matrix. It is shown below that the eigenvectors of the two-}

sided eigenvalue problem in (1.6.22) will only be orthogonal in the metric B and adhere to the constraints, {w₍′_i)Bw₍i)=1}, if they are calculated from the set of

orthonormal eigenvectors of the matrix B−1/2_AB−1/2_{, that is if the column vectors}

of U=B1/2_{W are orthonormal:}

W′BW=I

←→U′B−1/2BB−1/2U=I ←→U′U=I.