11. Hessenberg matrices

11.1 General

A Hessenberg matrix is a matrix such that all terms below its subdiagonal are zero. It is said to be H-regular if all terms in its subdiagonal are non-zero. It is obviously triangular (upper) if all the coefficients of its subdiagonal are zero.

The practical value of Hessenberg matrices is twofold:

• First, their determinants, like their characteristic polynomials, are easy to calculate;

• above all, any matrix is (effectively and whatever the base body) similar to a Hessenberg matrix.

This last point is established by simultaneous elementary operations (i.e. in such a way as to remain in the same similarity class) on rows and columns:...