Mathematic algorithm for basis
In mathematics, the Zassenhaus algorithm
is a method to calculate a basis for the intersection and sum of two subspaces of a vector space.
It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems.
Algorithm
Input
Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets:
and
Finally, let be linearly independent vectors so that and can be written as
and
Output
The algorithm computes the base of the sum and a base of the intersection .
Algorithm
The algorithm creates the following block matrix of size :
Using elementary row operations, this matrix is transformed to the row echelon form. Then, it has the following shape:
Here, stands for arbitrary numbers, and the vectors
for every and for every are nonzero.
Then with
is a basis of
and with
is a basis of .
Proof of correctness
First, we define to be the projection to the first component.
Let
Then and
.
Also, is the kernel of , the projection restricted to H.
Therefore, .
The Zassenhaus algorithm calculates a basis of H. In the first m columns of this matrix, there is a basis of .
The rows of the form (with ) are obviously in . Because the matrix is in row echelon form, they are also linearly independent.
All rows which are different from zero ( and ) are a basis of H, so there are such s. Therefore, the s form a basis of .
Example
Consider the two subspaces and of the vector space .
Using the standard basis, we create the following matrix of dimension :
Using elementary row operations, we transform this matrix into the following matrix:
- (Some entries have been replaced by "" because they are irrelevant to the result.)
Therefore
is a basis of , and
is a basis of .
See also
References
- Luks, Eugene M.; Rákóczi, Ferenc; Wright, Charles R. B. (April 1997), "Some algorithms for nilpotent permutation groups", Journal of Symbolic Computation, 23 (4): 335–354, doi:10.1006/jsco.1996.0092.
- Fischer, Gerd (2012), Lernbuch Lineare Algebra und Analytische Geometrie (in German), Vieweg+Teubner, pp. 207–210, doi:10.1007/978-3-8348-2379-3, ISBN 978-3-8348-2378-6
- The GAP Group (February 13, 2015), "24 Matrices", GAP Reference Manual, Release 4.7, retrieved 2015-06-11
External links
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