Proving Invertibility of Matrices: A Small Difference Makes a Big Impact

[Try hard]

For an n*n matrix C=(cij) over R or C, we define v(C)=Max|cij|

a.Show that if A is invertible, then B is invertible if v(A-B) is sufficiently small.

b. Show that for any, not necessarily invertible, n*n matix A, there is a sequence Ak of invertible matrices with v(A - Ak) -> 0 .

 

[matt grime]

If v(X)<e, it appears that a result along the lines of det(XY) < e^n * det(Y) is true. If so, you can use this for part a.

part b looks a little harder.

does this v define a (metric) topology on the nxn matrices?

 

[Try hard]

thanks for the hint on part a, the v in part b is the same as in part a