Smith normal form

Let $A$ be a non-zero $m cross times n$ matrix over a principal ideal domain (PID). Then, there exist invertible $m cross times m$ and $n cross times n$ matrices $S$ and $T$ , so that

$S times A times T space equals space open parentheses table row cell alpha subscript 1 end cell 0 0 blank midline horizontal ellipsis blank 0 row 0 cell alpha subscript 2 end cell 0 blank blank blank 0 row 0 0 down right diagonal ellipsis blank blank blank 0 row vertical ellipsis blank blank cell alpha subscript r end cell blank blank vertical ellipsis row blank blank blank blank 0 blank blank row blank blank blank blank blank down right diagonal ellipsis blank row 0 blank blank midline horizontal ellipsis blank blank 0 end table close parentheses$

where the diagonal elements satisfy $alpha subscript i vertical line alpha subscript i plus 1 end subscript$ . They are called invariant factors.

smith_normal_form(Matrix)

Description

smith_normal_form(Matrix)

Given a matrix, returns a vector with the invariant factors.

Smith normal form basis

Smith normal form

Syntax

Description

smith_normal_form(Matrix)

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