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LinearAlgebra[Generic][HessenbergForm] - compute the Hessenberg form of a square Matrix
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Calling Sequence
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HessenbergForm[F](A)
HessenbergForm[F](A,output=out)
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Parameters
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F
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a table or module, the domain of computation, a field
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A
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square Matrix of values in F
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out
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one of H, U or a list containing one or more of these names
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Description
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HessenbergForm[F](A) returns the upper Hessenberg form H of A.
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Given an n x n Matrix A of elements in a field F, the algorithm converts a copy of A into upper Hessenberg form H using O(n^3) operations in F. The algorithm requires that F be a field and should only be used if F is finite as there is severe expression swell in computing H.
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The (indexed) parameter F, which specifies the domain of computation, a field, must be a Maple table/module which has the following values/exports:
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F[`0`]: a constant for the zero of the ring F
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F[`1`]: a constant for the (multiplicative) identity of F
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F[`+`]: a procedure for adding elements of F (nary)
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F[`-`]: a procedure for negating and subtracting elements of F (unary and binary)
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F[`*`]: a procedure for multiplying two elements of F (commutative)
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F[`/`]: a procedure for dividing two elements of F
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F[`=`]: a boolean procedure for testing if two elements in F are equal
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Examples
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![with(LinearAlgebra[Generic])](/support/helpjp/helpview.aspx?si=4538/file06052/math130.png)
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![Q[`0`], Q[`1`], Q[`+`], Q[`-`], Q[`*`], Q[`/`], Q[`=`] := 0, 1, `+`, `-`, `*`, `/`, `=`](/support/helpjp/helpview.aspx?si=4538/file06052/math134.png)
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![A := Matrix([[2, -7, -3, 4], [1, -3, -4, 5], [-7, 10, 5, -7], [-7, 10, 5, -7]])](/support/helpjp/helpview.aspx?si=4538/file06052/math138.png)
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![A := Matrix([[2, -7, -3, 4], [1, -3, -4, 5], [-7, 10, 5, -7], [-7, 10, 5, -7]])](/support/helpjp/helpview.aspx?si=4538/file06052/math141.png)
| (1) |
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](/support/helpjp/helpview.aspx?si=4538/file06052/math145.png)
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![H := Matrix([[2, -14, 1, 4], [1, -10, 1, 5], [0, -46, 5, 28], [0, 0, 0, 0]])](/support/helpjp/helpview.aspx?si=4538/file06052/math148.png)
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](/support/helpjp/helpview.aspx?si=4538/file06052/math152.png)
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![H, U := Matrix([[2, -14, 1, 4], [1, -10, 1, 5], [0, -46, 5, 28], [0, 0, 0, 0]]), Matrix([[1, 0, 0, 0], [0, 1, 0, 0], [0, 7, 1, 0], [0, 0, -1, 1]])](/support/helpjp/helpview.aspx?si=4538/file06052/math155.png)
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, MatrixInverse[Q](U))](/support/helpjp/helpview.aspx?si=4538/file06052/math159.png)
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![Matrix([[2, -14, 1, 4], [1, -10, 1, 5], [0, -46, 5, 28], [0, 0, 0, 0]])](/support/helpjp/helpview.aspx?si=4538/file06052/math162.png)
| (4) |
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