invertible matrices are dense in set of nxn matrices
This is easily proven as follows. Take any invertible matrix (e.g. ), and consider the function (for or )
Clearly, is a polynomial function. It is not identically zero, for . But a non-zero polynomial has only finitely many zeroes, So given any single point , if is close enough but unequal to , must be non-zero. In particular, applying this for , we see that the matrix is invertible for small . And the distance of this matrix from is , which becomes small as gets small.
|Title||invertible matrices are dense in set of nxn matrices|
|Date of creation||2013-03-22 15:38:51|
|Last modified on||2013-03-22 15:38:51|
|Last modified by||stevecheng (10074)|