If a matrix has all elements in the lower triangular and the superdiagonal positions are non zero and zero otherwise, then can I name it?
To be precise, they called lower Hessenberg matrices. Similarly one can define upper
Hessenberg matrices.
This is a lower Hessenberg matrix.
An upper Hessenberg has the structure of the transpose of the preceding one that is, zero entries below the diagonal below and next to the main diagonal (subdiagonal).
Symmetric Hessenberg matrices are tridiagonal with only three non zero diagonals.
Dear Professor Meurant,
Can we calculate the computational time for finding the determinant of Hessenberg matrices ?
If you want to compute the determinant of your Hessenberg matrix, there are several ways.
You can easily write a recurrence for the determinant of a Hessenberg matrix.
Numerically you can compute the LU factorisation of H or H^T and you get the determinants with the products of the diagonal entries.
Or you can compute a QR factorization of H^T. This is easy to do with Givens rotations.
Then you have H^T = QR and det(H)=det(R)= product of the diagonal entries of R.
Or you can use Hyman's method see N.J. Higham, Accuracy and stability of Numerical algorithms, SIAM 2002, p. 280.
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