eigenvalues λ and associated eigenvectors

The

The method is not suitable for determining complex eigenvalues. However, these do not occur at all with symmetric matrices, for example.

With the help of Gerschgorin circles, the position of the eigenvalues is estimated in order to determine suitable spectral shifts.

The eigenvalue found in each case and the Gerschgorin circles for eigenvalue estimation are displayed in the complex plane.

If you want to determine eigenvalues that have no extremal position, you can use the

If you make a spectral shift by

which was originally closest to

We start with a vector

If the method converges,

The largest eigenvalue can then be determined using the so-called Rayleigh quotient:

λ

So you always just have to multiply the matrix with the last approximation and then normalize the result vector.

If the difference between two approximations is small enough, you stop.

The eigenvalues of the inverse

When analyzing the eigenvalues of

The largest magnitude eigenvalue of

and the smallest magnitude eigenvalue of

Consequently, the power method can also be used to determine the

You only have to do the iteration with the inverse of the respective matrix and take the reciprocal of the eigenvalue found.

then the matrix

Therefore, all eigenvalues shift by the size c.

The eigenvectors do

Thus, to determine an eigenvalue that is assumed to be close to c,

first create a matrix with a spectral shift of -c and then using the inverse power method

determine the smallest eigenvalue λ

The eigenvalue of the original matrix you are looking for is then λ

in whose union all eigenvalues of a matrix lie.

The circle centers are the diagonal elements of the matrix.

The radii of the circles are determined from the sum of the amounts of the remaining row elements.

Alternatively, you can also add up the amounts of the remaining column elements.

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