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MatrixPolynomialAlgebra[HermiteForm] - compute the Hermite normal form of a Matrix (in row or column form)
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Calling Sequence
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HermiteForm(A, x, out)
HermiteForm[row](A, x, out)
HermiteForm[column](A, x, out)
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Parameters
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A
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Matrix
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x
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name; variable name of the polynomial domain
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out
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(optional) equation of the form output = obj where obj is one of 'H' or 'U', or a list containing one or more of these names; select result objects to compute
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Description
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The HermiteForm(A, x) and HermiteForm[row](A, x) commands compute the Hermite normal form (row-reduced echelon form) of an m x n rectangular Matrix of univariate polynomials in x over the field of rational numbers Q, or rational expressions over Q, that is, univariate polynomials in x with coefficients in Q(a1,...,an).
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The HermiteForm[column](A, x) command computes the Hermite normal form (column-reduced echelon form) of A.
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The row (column) Hermite normal form is obtained by performing elementary row (column) operations on A. This includes interchanging rows (columns), multiplying a row (column) by a unit, and subtracting a polynomial multiple of one row (column) from another.
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The number of nonzero rows (columns) of the Hermite Form, H, is the rank of A. If n = m, then ![product(H[i, i], i = 1 .. n) = normal(Determinant(A))](/support/helpjp/helpview.aspx?si=5715/file02406/math118.png) where normal means unit normal, that is, monic.
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Option
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The output option (out) determines the content of the returned expression sequence.
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As determined by the out option, an expression sequence containing one or more of the factors H (the Hermite normal form) or U (the transformation Matrix) is returned. If obj is a list, the objects are returned in the order specified in the list.
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The returned Matrix objects have the property that  for row Hermite normal form, and  for column Hermite normal form.
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Examples
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![A := Matrix([[3+x, 1, -4], [4, x, 2], [x^2-1, 4, -1]])](/support/helpjp/helpview.aspx?si=5715/file02406/math158.png)
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![H := Matrix([[1, 0, -53/152-(7/76)*x^2+(29/304)*x], [0, 1, (31/76)*x+(3/19)*x^2-37/38], [0, 0, x^3+(1/4)*x^2-(15/4)*x-43/2]])](/support/helpjp/helpview.aspx?si=5715/file02406/math165.png)
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](/support/helpjp/helpview.aspx?si=5715/file02406/math169.png)
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![H, U := Matrix([[1, 0, -53/152-(7/76)*x^2+(29/304)*x], [0, 1, (31/76)*x+(3/19)*x^2-37/38], [0, 0, x^3+(1/4)*x^2-(15/4)*x-43/2]]), Matrix([[(7/304)*x^2-(9/304)*x+3/19, -(7/304)*x+37/304, -(7/304)*x-3/76], [-(3/76)*x^2-(7/76)*x+3/19, (3/76)*x-5/76, (3/76)*x+4/19], [-(1/4)*x^3+(1/4)*x+4, (1/4)*x^2-x-13/4, (1/4)*x^2+(3/4)*x-1]])](/support/helpjp/helpview.aspx?si=5715/file02406/math172.png)
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![Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])](/support/helpjp/helpview.aspx?si=5715/file02406/math179.png)
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![mul(H[i, i], i = 1 .. 3)](/support/helpjp/helpview.aspx?si=5715/file02406/math183.png)
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](/support/helpjp/helpview.aspx?si=5715/file02406/math190.png)
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, `<,>`(0, 2*y, x*y)), x)](/support/helpjp/helpview.aspx?si=5715/file02406/math204.png)
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![Matrix([[1, y], [0, 0], [0, 0]])](/support/helpjp/helpview.aspx?si=5715/file02406/math207.png)
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](/support/helpjp/helpview.aspx?si=5715/file02406/math211.png)
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![H, U := Matrix([[1, 0, 0], [0, 1, 0], [-2/21-(1/42)*x+(2/21)*x^2, -29/42+(4/21)*x^2-(1/21)*x, x^3+(1/4)*x^2-(15/4)*x-43/2]]), Matrix([[2/21, 4/21, 1/2+x], [-1/21, -2/21, -(1/2)*x-11/2], [(1/42)*x-4/21, (1/21)*x+5/42, (1/4)*x^2+(3/4)*x-1]])](/support/helpjp/helpview.aspx?si=5715/file02406/math214.png)
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![Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]])](/support/helpjp/helpview.aspx?si=5715/file02406/math221.png)
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![mul(H[i, i], i = 1 .. 3)](/support/helpjp/helpview.aspx?si=5715/file02406/math225.png)
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](/support/helpjp/helpview.aspx?si=5715/file02406/math232.png)
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, `<,>`(0, 2*y, x*y)), x)](/support/helpjp/helpview.aspx?si=5715/file02406/math246.png)
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![Matrix([[0, 0], [1, 0], [(1/2)*x, 0]])](/support/helpjp/helpview.aspx?si=5715/file02406/math249.png)
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