Difference between revisions of "Manuals/calci/HESSENBERG"
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*Here MATRIX("hessenberg") displays the hessenberg matrix of order 3. | *Here MATRIX("hessenberg") displays the hessenberg matrix of order 3. | ||
*An nxn matrix with a_i,j=0 for i>j+1 is called a Hessenberg matrix. | *An nxn matrix with a_i,j=0 for i>j+1 is called a Hessenberg matrix. | ||
| − | *So the form of a Hessenberg matrix is: | + | *So the form of a Hessenberg matrix is: \begin{bmatrix} |
| − | |||
| − | |||
| − | |||
| − | \begin{bmatrix} | ||
a_{11} & a_{12} & a_{13} \cdots & a_{1(n-1)}& a_{1n} \\ | 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} \\ | a_{21} & a_{22} & a_{23} \cdots& a_{2(n-1)}& a_{2n} \\ | ||
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0 & 0 & 0 & a_{(n-1)(n-2)} & a_{(n-1)(n-1)} & a_{(n-1)n}\\ | 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}\\ | 0 & 0 & 0 & 0 & a_{n(n-1)} & a_{nn}\\ | ||
| − | \end{bmatrix} | + | \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") | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | | -10 || -50 || -92 | ||
| + | |- | ||
| + | | -32 || 62 || 33 | ||
| + | |- | ||
| + | | 0 || -99 || -81 | ||
| + | |} | ||
| + | *2.MATRIX("hessenberg",6) | ||
| + | {| class="wikitable" | ||
| + | |- | ||
| + | | 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== | ||
| + | *[[Manuals/calci/HADAMARD| HADAMARD]] | ||
| + | *[[Manuals/calci/CONFERENCE| CONFERENCE]] | ||
| + | *[[Manuals/calci/CIRCULANT| CIRCULANT]] | ||
| + | *[[Manuals/calci/HANKEL| HANKEL]] | ||
| + | |||
| + | ==References== | ||
Revision as of 13:08, 24 April 2015
MATRIX("HESSENBERG",order)
- is the order of the Hessenberg matrix.
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 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 |