NMax=1000;

A=Table[1/(i+j+1),
{i,0.,NMax-1},
{j,0.,NMax-1}];
Dimensions[A]
Timing[Eigenvalues[A];]

Timing[
NMax=1000;

A=Table[1/(i+j+1),
{i,0.,NMax-1},
{j,0.,NMax-1}];
Dimensions[A]
Timing[EVals=Eigenvalues[A];]
]
Take[EVals,4]

[struther@gossamer MatrixExample4] $ time example
eigenvalue = 2.44315
eigenvalue = 1.24364
eigenvalue = 0.492524
eigenvalue = 0.173155
eigenvalue = 0.0571838
eigenvalue = 0.018108
30.457 u 0.130 s 0 : 30.90 98.9 % 0 + 0 k 0 + 0 io 0 pf + 0 w
[struther@gossamer MatrixExample4] $