# Manuals/calci/HESSENBERG

MATRIX("HESSENBERG",order)

• is the order of the Hessenberg matrix.

## Description

• This function gives the matrix with the property of Hessenberg matrix.
• 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.
• Here MATRIX("hessenberg") displays the hessenberg matrix of order 3.
• An nxn matrix with for is called a Hessenberg matrix.
• So the form of a Hessenberg matrix is: \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} \\ 0 & 0 & 0 \cdots& a_{5(n-1)}& a_{5n} \\ \vdots & \ddots & \vdots \\ 0 & 0 & 0 & a_{(n-1)(n-2)} & a_{(n-1)(n-1)} & a_{(n-1)n}\\ 0 & 0 & 0 & 0 & a_{n(n-1)} & a_{nn}\\ \end{bmatrix}

• So the matrix is zero below the first subdiagonal.
• If the matrix is symmetric or Hermitian, the form is tridiagonal.

## Examples

• 1.MATRIX("hessenberg") =53
• 2.MATRIX("hessenberg",3)
 -10 -50 -92 -32 62 33 0 -99 -81
• 3.MATRIX("hessenberg",6)
 99 88 -73 20 -17 -10 -28 40 -2 15 -48 55 0 -46 56 -76 -85 -70 0 0 12 -72 72 -17 0 0 0 -67 -26 -6 0 0 0 0 -13 50

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Hessenberg Matrix