|
linalg[ismith] - integer-only Smith normal form
|
|
Calling Sequence
|
|
ismith(A)
ismith(A, U, V)
|
|
|
Parameters
|
|
|
A
|
-
|
rectangular matrix of integers
|
|
U
|
-
|
name
|
|
V
|
-
|
name
|
|
|
|
|
|
Description
|
|
|
•
|
The function ismith computes the Smith normal form S of an n by m rectangular matrix of integers.
|
|
•
|
If two n by n matrices have the same Smith normal form, they are equivalent.
|
|
•
|
The Smith normal form is a diagonal matrix where
|
|
rank(A) = number of nonzero rows (columns) of S
|
|
sign(S[i,i]) = 1 for 0 < i <= rank(A)
|
|
S[i,i] divides S[i+1,i+1] for 0 < i < rank(A)
|
|
product(S[i,i],i=1..r) divides det(M) for all minors M of rank 0 < r <= rank(A)
|
|
|
|
|
Hence if n = m and rank(A) = n then ![abs(det(A)) = product(S[i, i], i = 1 .. n)](/support/helpjp/helpview.aspx?si=6216/file07328/math113.png) .
|
|
•
|
The Smith normal form is obtained by doing elementary row and column operations. This includes interchanging rows (columns), multiplying through a row (column) by -1, and adding integral multiples of one row (column) to another.
|
|
•
|
Although the rank and determinant can be easily obtained from S this is not an efficient method for computing these quantities except that this may yield a partial factorization of  without doing any explicit factorizations.
|
|
•
|
In the case of three arguments, the second argument U and the third argument V will be assigned the transformation matrices on output, such that smith(A) = U &* A &* V.
|
|
•
|
The command with(linalg,ismith) allows the use of the abbreviated form of this command.
|
|
|
Download Help Document
Was this information helpful?
