Hessenberg matrix
HessenbergMatrix
2002-01-14 02:36:46
2006-10-28 23:08:30
Definition
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An \emph{upper Hessenberg matrix} is of the form
$$ \begin{bmatrix}
a_{11} & a_{12} & a_{13} & \cdots & a_{1,n-1} & a_{1n} \\
a_{21} & a_{22} & a_{23} & \cdots & a_{2,n-1} & a_{2n} \\
0 & a_{32} & a_{33} & \cdots & a_{3,n-1} & a_{3n} \\
0 & 0 & a_{43} & \cdots & a_{4,n-1} & a_{4n} \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & 0 & a_{n,n-1} & a_{nn}
\end{bmatrix} $$
and a \emph{lower Hessenberg matrix} is of the form
$$ \begin{bmatrix}
a_{11} & a_{12} & 0 & \cdots & 0 & 0 \\
a_{21} & a_{22} & a_{23} & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
a_{n-2,1} & a_{n-2,2} & a_{n-2,3} & \cdots & a_{n-2,n-1} & 0 \\
a_{n-1,1} & a_{n-1,2} & a_{n-1,3} & \cdots & a_{n-1,n-1} & a_{n-1,n} \\
a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n-1} & a_{n,n}
\end{bmatrix} $$