invertible matrices are dense in set of nxn matrices

If A is any n×n matrix with real or complex entries, Then there are invertible matrices arbitrarily close to A, under any norm for the n×n matrices.

This is easily proven as follows. Take any invertible matrix B (e.g. B=I), and consider the function (for t or )


Clearly, p is a polynomial function. It is not identically zero, for p(1)=detB0. But a non-zero polynomialPlanetmathPlanetmath has only finitely many zeroes, So given any single point t0, if t is close enough but unequal to t0, p(t) must be non-zero. In particular, applying this for t0=0, we see that the matrix (1-t)A+tB is invertiblePlanetmathPlanetmathPlanetmath for small t0. And the distance of this matrix from A is |t|B-A, which becomes small as t gets small.

Title invertible matrices are dense in set of nxn matrices
Canonical name InvertibleMatricesAreDenseInSetOfNxnMatrices
Date of creation 2013-03-22 15:38:51
Last modified on 2013-03-22 15:38:51
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 5
Author stevecheng (10074)
Entry type Theorem
Classification msc 15A09