Matrices with large condition numbers can be seen as nearly singular matrices. The MATLAB experiment below illustrates the relation with eigenvalues.
>> % We can see condition number as the ratio of the largest
>> % and smallest eigenvalue and use this to make matrices of
>> % with any condition number we want.
>> Q = orth(rand(5)); % eigenvectors
>> D = diag(rand(1,5)) % eigenvalues
D =
0.2028 0 0 0 0
0 0.1987 0 0 0
0 0 0.6038 0 0
0 0 0 0.2722 0
0 0 0 0 0.1988
>> D(1,1) = eps
D =
0.0000 0 0 0 0
0 0.1987 0 0 0
0 0 0.6038 0 0
0 0 0 0.2722 0
0 0 0 0 0.1988
>> A = Q*D*Q'
A =
0.1750 -0.0652 -0.0282 -0.0421 -0.0440
-0.0652 0.2421 -0.0782 0.0439 -0.1332
-0.0282 -0.0782 0.1664 -0.0571 -0.0340
-0.0421 0.0439 -0.0571 0.2783 -0.1948
-0.0440 -0.1332 -0.0340 -0.1948 0.4117
>> cond(A)
ans =
3.2068e+15
>> % This condition number is about 1/eps.