proportions of invertible matrices
Let denote the invertible -matrices over a ring , and the set of all -matrices over . When is a finite field of order , commonly denoted or , we prefer to write simply . In particular, is a power of a prime.
The number of -matrices over a is . When a matrix is invertible, its rows form a basis of the vector space and this leads to the following formula
(Refer to order of the general linear group. (http://planetmath.org/OrdersAndStructureOfClassicalGroups))
Now we prove the ratio holds:
As and is fixed, the proportion of invertible matrices decreases monotonically and converges towards a positive limit. Furthermore,
By direct expansion we find
So setting and
for all , we have
As , for all and as , we may use Leibniz’s theorem to conclude the alternating series converges. Furthermore, we may estimate the error to the -th term with error within . Using we have an estimate of with error . Since this gives with error . Thus we have at least chance of choosing an invertible matrix at random. ∎
is the only field size where the proportion of invertible matrices to all matrices is less than .
Acknowledgements: due to discussions with Wei Zhou, silverfish and mathcam.
|Title||proportions of invertible matrices|
|Date of creation||2013-03-22 15:57:35|
|Last modified on||2013-03-22 15:57:35|
|Last modified by||Algeboy (12884)|
|Synonym||proportions invertible linear transformations|