compute the Hermite Normal Form of a matrix mod p
compute the Smith Normal Form of a matrix mod p
Hermite(A, x, U)
Smith(A, x, U, V)
rectangular matrix of univariate polynomials in x
Hermite and Smith are placeholders for representing the Hermite and Smith Normal Forms respectively. They are used in conjunction with mod as is described below.
Hermite(A, x) mod p computes the Hermite Normal Form (reduced row echelon form) of an m by n rectangular matrix of univariate polynomials in x over the integers modulo p. The polynomial coefficients must be rational or elements of a finite extension field specified by RootOfs. In the case of three arguments, the third argument, U, will be assigned the transformation matrix upon completion, such that Hermite(A) = U &* A.
Smith(A, x) mod p computes the Smith Normal Form of a matrix with univariate polynomial entries in x over the integers modulo p. The coefficients of the polynomial must be either rational or elements of a finite extension field specified by RootOfs. In the case of four arguments, the third argument U and the fourth argument V will be assigned the transformation matrices on output, such that Smith(A) = U &* A &* V.
A ≔ Matrix⁡1+x,1+x2,1+x2,1+x4
A ≔ Matrix⁡1+x,1+x2,1+x2,1+x4:Hermite⁡A,x,Umod2
To obtain the column form for the HNF of A do
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