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Definition of Hessenberg form

Created over 8 years ago, updated 10 days ago

Definition: A square matrix $H$ is in upper Hessenberg form if all entries below the first subdiagonal are zero: $h_{ij} = 0$ for $i > j + 1$.

$$H = \begin{pmatrix} * & * & * & * & * \\ * & * & * & * & * \\ 0 & * & * & * & * \\ 0 & 0 & * & * & * \\ 0 & 0 & 0 & * & * \end{pmatrix}$$

Similarly, lower Hessenberg has zeros above the first superdiagonal.

Theorem: Every square matrix is unitarily similar to a matrix in Hessenberg form: $A = QHQ^*$ where $Q$ is unitary.

Hessenberg form is used as an intermediate step in eigenvalue algorithms (like the QR algorithm) because it is nearly triangular and cheaper to work with than a full matrix. The reduction to Hessenberg form can be done in $O(n^3)$ operations using Householder reflections.