HomeMatlabWhat Is a Hessenberg Matrix? – Nick Higham

What Is a Hessenberg Matrix? – Nick Higham

December 5, 2023

A well-known Hessenberg matrix with some attention-grabbing properties is the Frank Matrix. For instance, the eigenvalues seem in reciprocal pairs $(lambda,1/lambda)$ . For $n = 5$ :

>> F = gallery('frank',5)
F =
     5     4     3     2     1
     4     4     3     2     1
     0     3     3     2     1
     0     0     2     2     1
     0     0     0     1     1

>> e = eig(F); type([e 1./e])
ans =
   9.9375e-02   9.9375e-02
   2.8117e-01   2.8117e-01
   1.0000e+00   1.0000e+00
   3.5566e+00   3.5566e+00
   1.0063e+01   1.0063e+01

One other well-known Hessenberg matrix is the Grcar matrix, a Toeplitz matrix of the shape

>> G = gallery('grcar',5)
G =
     1     1     1     1     0
    -1     1     1     1     1
     0    -1     1     1     1
     0     0    -1     1     1
     0     0     0    -1     1

It has attention-grabbing pseudospectra.

Companion matrices are higher Hessenberg with a unit subdiagonal:

$notag C = begin{bmatrix} a_{n-1} & a_{n-2} & dots &dots & a_0 1 & 0 & dots &dots & 0 0 & 1 & ddots & & vdots vdots & & ddots & 0 & 0 0 & dots & dots & 1 & 0 end{bmatrix}.$