A 2 x 2 generalized eigenvalue problem

The generalized eigenvalue problem is unfortunately considerably more complicated than its regular counterpart when S = I. There are possibilities for accidental cases when basis functions apparently should mix, but they do not. We can give a simple example of this for a 2 x 2 system. Assume we have the pair of matrices 

It is not difficult to show that the eigenvectors of H are the same as those of H. 

Our generalized eigenvalue problem thus depends upon three parameters, a, b, and s. Denoting the eigenvalue by W and solving the quadratic equation, we obtain 

We note the possibility ofan accident that cannot happen if 5 = 0 and ft 0 Should b = -Inas, one of the two values of W is either a, and one of the two diagonal elements of H is unchanged. Let us for dehniteness assume that b = as and it is a we obtain. Then, clearly the vector Ci we obtain is 

We do not discuss it, but there is an x version of this complication. If there is no degeneracy, one of the diagonal elements of the //-matrix may be unchanged in going to the eigenvalues, and the eigenvector associated with it is [0,. ..,0, 1, 0, 

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