Manuals/calci/HESSENBERG

MATRIX("HESSENBERG",order)

• $order$ 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 $a_{i,j}=0$ for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\> j+1} 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")

-10	-50	-92
-32	62	33
0	-99	-81

• 2.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

See Also

• HADAMARD
• CONFERENCE
• CIRCULANT
• HANKEL