In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg.

A Hessenberg decomposition is a matrix decomposition of a matrix ${\displaystyle A}$ into a unitary matrix Failed to parse (unknown error): {\displaystyle P} and a Hessenberg matrix ${\displaystyle H}$ such that $${\displaystyle PHP^{*}=A}$$ where ${\displaystyle P^{*}}$ denotes the conjugate transpose.

A square ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if ${\displaystyle a_{i,j}=0}$ for all ${\displaystyle i,j}$ with ${\displaystyle i>j+1}$.

An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if ${\displaystyle a_{i+1,i}\neq 0}$ for all ${\displaystyle i\in \{1,\ldots ,n-1\}}$.

A square ${\displaystyle n\times n}$ matrix ${\displaystyle A}$ is said to be in lower Hessenberg form or to be a lower Hessenberg matrix if its transpose ${\displaystyle }$ is an upper Hessenberg matrix or equivalently if ${\displaystyle a_{i,j}=0}$ for all ${\displaystyle i,j}$ with ${\displaystyle j>i+1}$.

A lower Hessenberg matrix is called unreduced if all superdiagonal entries are nonzero, i.e. if ${\displaystyle a_{i,i+1}\neq 0}$ for all ${\displaystyle i\in \{1,\ldots ,n-1\}}$.

Consider the following matrices. $${\displaystyle A={\begin{bmatrix}1&4&2&3\\3&4&1&7\\0&2&3&4\\0&0&1&3\\\end{bmatrix}}}$$ $${\displaystyle B={\begin{bmatrix}1&2&0&0\\5&2&3&0\\3&4&3&7\\5&6&1&1\\\end{bmatrix}}}$$ $${\displaystyle C={\begin{bmatrix}1&2&0&0\\5&2&0&0\\3&4&3&7\\5&6&1&1\\\end{bmatrix}}}$$

The matrix ${\displaystyle A}$ is an upper unreduced Hessenberg matrix, ${\displaystyle B}$ is a lower unreduced Hessenberg matrix and ${\displaystyle C}$ is a lower Hessenberg matrix but is not unreduced.