eigenvalues λ and associated eigenvectors x of the matrix A are calculated iteratively.
The power method goes back to Richard von Mises and Helmut Wielandt.
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 inverse power method with spectral shift.
If you make a spectral shift by -v, all eigenvalues of the matrix shift in such a way that the eigenvalue,
which was originally closest to +v, becomes the absolute smallest and can therefore be found using the inverse power method.
Power Method
To determine the eigenvector of the largest magnitude eigenvalue of a matrix A you can use the following algorithm:
xn = Axn-1 / |Axn-1|
We start with a vector x0, which contains random numbers.
If the method converges, xn converges against the eigenvector to the eigenvalue with the largest magnitude.
The largest magnitude eigenvalue can then be determined using the so-called Rayleigh quotient:
λmax = xTAx / (xTx)
So you have to multiply the matrix again and again with the last approximation and normalize the result vector.
If the difference between two approximations is small enough, you stop.
Inverse Power Method
The eigenvectors of the inverse A-1 of a matrix are the same as those of the matrix A.
The eigenvalues of the inverse A-1 are the reciprocals of the eigenvalues of A.
When analyzing the eigenvalues of A, one can therefore also start from the inverse A-1.
The largest magnitude eigenvalue of A now corresponds to the smallest magnitude one of A-1
and the smallest magnitude eigenvalue of A corresponds to the largest magnitude one of A-1.
Consequently, the power method can also be used to determine the smallest magnitude eigenvalue and the associated eigenvector of a matrix.
You only have to do the iteration with the inverse of the respective matrix and finally take the reciprocal of the eigenvalue found.
Spectral shift
If a matrix A has the eigenvalues λ1, λ2, λ3, ... ,
then the matrix A - cI has the eigenvalues λ1-c, λ2-c, λ3-c, ...
Therefore, all eigenvalues shift by the size c.
The eigenvectors do not change with this spectral shift.
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 λS of this shifted matrix.
The eigenvalue of the original matrix you are looking for is then λS+c.
Gerschgorin circles
According to the eigenvalue estimation of Gerschgorin, there are circular disks in the complex number plane,
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.