65-00-HessenbergMatrix.tex

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\pmtitle{Hessenberg matrix}
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\pmsynonym{upper Hessenberg matrix}{HessenbergMatrix}
\pmsynonym{lower Hessenberg matrix}{HessenbergMatrix}
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\pmsynonym{lower Hessenberg}{HessenbergMatrix}

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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

$$ \begin{bmatrix}
a_{11} & a_{12} & 0      & \cdots &   0       &  0     \\
a_{21} & a_{22} & a_{23} & \cdots &   0       &  0     \\
\vdots & \vdots & \vdots & \ddots & \vdots    & \vdots \\
a_{n-2,1} & a_{n-2,2} & a_{n-2,3} & \cdots & a_{n-2,n-1} & 0 \\
a_{n-1,1} & a_{n-1,2} & a_{n-1,3} & \cdots & a_{n-1,n-1} & a_{n-1,n} \\
a_{n,1} & a_{n,2} & a_{n,3} & \cdots & a_{n,n-1} & a_{n,n} 
\end{bmatrix} $$
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