The inverse eigenvalue problem

Now assume to be symmetric with non-zero off-diagonal elements, i.e. , . Let be the eigenvalues and the normalized eigenvectors of T, i.e.

We consider the inverse problem: Determine from and . It is well known that this problem can easily be solved by the Lanczos method.

The Lanczos algorithm computes for a symmetric matrix A a symmetric tridiagonal matrix T and a unitary matrix U such that

 

in the following way. With the rows of U, (3.1) reads

Now let be arbitrary. Then, from equation 1,

Thus , , , are determined. Likewise, from equation 2,

yielding , . Proceeding in this fashion in equations up to and including n-1 we obtain , and . is obtained from equation n.

In order to solve our inverse problem we simply apply the Lanczos method to the matrix . The same problem can be solved by the Gelfand-Levitan method in the following way.

Let be an arbitrary known symmetric tridiagonal matrix, subject only to the condition , . We introduce solutions , of

and correspondingly for using instead of T. In other words, , satisfy

 

 

Note that for , is an eigenvector of T, hence

 

i.e. the projection operator E can be dropped in that case.

Proof: Let . Then, because of (2.1), (3.4),

Since , . Thus and both satisfy (3.3) with . Because of , (3.3) is uniquely solvable. Hence .

Now we come to the core of the Gelfand-Levitan method. We introduce the matrices

Assume that , . Then, , hence and

Since ,

 

Theorem 3.1 means that

 

Inserting this into (3.5) yields

Since , P are known from the data, L can be computed by a Cholesky decomposition of . Once L is determined, T can be computed from

This solves the inverse eigenvalue problem.