Page "Characteristic polynomial" Paragraph 53

The characteristic polynomial is defined by the determinant of the matrix with a shift.

It has zeros only, without any pole.

Commonly, the secular function implies the characteristic polynomial.

But, in the strict sense, the secular function has poles as well.

Interestingly, the poles are located in the eigenvalues of its sub-matrices.

Thus, if the information of the sub-matrices is available, the eigenvalues of the matrix can be described using that kind of information.

Furthermore, by partitioning the matrix like matrix tearing or gruing, we can iterate the eigenvalues in a recursive way.

According to the methods of partitioning, the variant forms of the secular functions can be built up.

However, they are all of the form of a series of the simple rational functions, which have poles at the eigenvalues of the partitioned matrices.

For example, we can find a form of secular function in the divide-and-conquer eigenvalue algorithm.