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Elementary Arithmetic

A + B : Mtrx, Mtrx -> Mtrx
Given m x n matrices A and B over a ring R, return A + B.
A - B : Mtrx, Mtrx -> Mtrx
Given m x n matrices A and B over a ring R, return A - B.
A * B : Mtrx, Mtrx -> Mtrx
Given an m x n matrix A over a ring R and an n x p matrix B over R, return the m x p matrix A.B over R. This function attempts to preserve the maximal amount of information in the choice of parent for the product. For example, if A and B are both square and have the same matrix algebra M as parent, then the product will also have M as parent. Similarly, if the parents of A and B are R-matrix spaces such that the codomain of B equals the domain A, then the product will have domain equal to that of A and codomain equal to that of B.
x * A : RngElt, Mtrx -> Mtrx
A * x : Mtrx, RngElt -> Mtrx
Given an m x n matrix A over a ring R and a ring element x coercible into R, return the scalar product x.A.
- A : Mtrx -> Mtrx
Given a matrix A, return -A.
A ^ -1 : Mtrx, RngIntElt -> Mtrx
Given a invertible square matrix A over a ring R, return the inverse B of A so that A.B = B.A = 1. The coefficient ring R must be either a field, a Euclidean domain, or a ring with an exact division algorithm and having characteristic equal to zero or greater than m (this includes most commutative rings).
A ^ n : Mtrx, RngIntElt -> Mtrx
Given a square matrix A over a ring R and an integer n, return the matrix power An. A0 is defined to be the identity matrix for any square matrix A (even if A is zero). If n is negative, A must be invertible (see the previous function), and the result is (A - 1) - n.
Transpose(A) : Mtrx -> Mtrx
Given an m x n matrix A over a ring R, return the transpose of A, which is simply the n x m matrix over R whose (i, j)-th entry is the (j, i)-th entry of A.
AddScaledMatrix(A, s, B) : Mtrx, RngElt, Mtrx -> Mtrx
Given a matrix A over a ring R, a scalar s coercible into R, and a matrix B over R with the same shape as A, return A + s.B. This is generally quicker than the call A + s*B.
AddScaledMatrix(~A, s, B) : Mtrx, RngElt, Mtrx ->
Given a matrix A over a ring R, a scalar s coercible into R, and a matrix B over R with the same shape as A, set A to A + s.B. This is generally quicker than the statement A := A + s*B;.
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013