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linalg[hermite] - Hermite Normal Form (reduced row echelon form)
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Calling Sequence
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hermite(A, x)
hermite(A, x, U)
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Parameters
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A
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rectangular matrix of polynomials in x
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x
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name
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U
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name
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Description
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The function hermite computes the Hermite Normal Form (reduced row echelon form) of an m by n rectangular matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q.
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In principle this should work for polynomials in x over any field F, i.e. the Euclidean domain F[x], but in practice the code is only as powerful as Maple's normal function.
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The Hermite normal form is obtained by doing elementary row operations on A. This includes interchanging rows, multiplying through a row by a unit, and subtracting a multiple of one row from another.
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One can use transposes to obtain the column form of the Hermite Normal Form of a matrix.
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In the case of three arguments, the third argument U will be assigned the transformation matrix on output, such that the following holds: hermite(A) = U &* A.
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The command with(linalg,hermite) allows the use of the abbreviated form of this command.
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Examples
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Important: The linalg package has been deprecated. Use the superseding packages, LinearAlgebra and VectorCalculus, instead.
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![H := matrix([[-(-3+x)^2*(-2+x), (-3+x)*(-2+x)*(-4+x)], [(-3+x)*(-2+x)*(-4+x), -(-3+x)^2*(-4+x)]])](/support/helpjp/helpview.aspx?si=6208/file07319/math142.png)
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![matrix([[6+x^2-5*x, 0], [0, 12+x^2-7*x]])](/support/helpjp/helpview.aspx?si=6208/file07319/math149.png)
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![H := matrix(3, 3, [[1, x, x^2], [x^3+2*x-1, 0, 1], [1, x-1, x^2+1]])](/support/helpjp/helpview.aspx?si=6208/file07319/math153.png)
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![H := matrix([[1, x, x^2], [x^3+2*x-1, 0, 1], [1, x-1, x^2+1]])](/support/helpjp/helpview.aspx?si=6208/file07319/math156.png)
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![matrix([[1, 0, x^2+x], [0, 1, -1], [0, 0, x^5+x^4+2*x^3+x^2-x-1]])](/support/helpjp/helpview.aspx?si=6208/file07319/math163.png)
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To obtain the column form of HNF for H do
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![matrix([[1, 0, 0], [0, 1, 0], [2-x^3-x-x^4-2*x^2, 1+x, x^5+x^4+2*x^3+x^2-x-1]])](/support/helpjp/helpview.aspx?si=6208/file07319/math172.png)
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![A := matrix(2, 3, [[x+1, x^2+1, 2*x-1], [x+2, x^2+x, x-2]])](/support/helpjp/helpview.aspx?si=6208/file07319/math176.png)
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![B := matrix([[1, 0, (-(1/2)*x^2-(1/2)*x)*(2*x-1)+((1/2)*x^2+1/2)*(-2+x)], [0, 1, ((1/2)*x+1)*(2*x-1)+(-1/2-(1/2)*x)*(-2+x)]])](/support/helpjp/helpview.aspx?si=6208/file07319/math183.png)
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![matrix([[-(1/2)*x^2-(1/2)*x, (1/2)*x^2+1/2], [(1/2)*x+1, -1/2-(1/2)*x]])](/support/helpjp/helpview.aspx?si=6208/file07319/math190.png)
| (7) |
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![matrix([[0, 0, 0], [0, 0, 0]])](/support/helpjp/helpview.aspx?si=6208/file07319/math197.png)
| (8) |
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