> A = [3,0,0;0,2,0;0,0,2]
A =
     3     0     0
     0     2     0
     0     0     2

> B = rand(3,3)
B =
    0.9501    0.4860    0.4565
    0.2311    0.8913    0.0185
    0.6068    0.7621    0.8214

> A = inv(B)*A*B
A =
    3.5906    0.8136    0.7641
   -0.3957    1.7976   -0.1901
   -0.8079   -0.4133    1.6118

> eig(A)
ans =
    3.0000
    2.0000
    2.0000

> format compact
> z = rand(3,1);
perform the power method to find the largest eigenvalue.
> for i=1:50, w = A*z; lambda(i) = w(1)/z(1); z = w/sqrt(sum(w.*w)); end
here are the first 50 approximate eigenvalues.  They're converging to 3. 
> lambda
lambda =
  Columns 1 through 7 
    6.0773    4.0127    3.5048    3.2880    3.1752    3.1104    3.0710
  Columns 8 through 14 
    3.0462    3.0303    3.0200    3.0133    3.0088    3.0059    3.0039
  Columns 15 through 21 
    3.0026    3.0017    3.0012    3.0008    3.0005    3.0003    3.0002
  Columns 22 through 28 
    3.0002    3.0001    3.0001    3.0000    3.0000    3.0000    3.0000
  Columns 29 through 35 
    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000
  Columns 36 through 42 
    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000
  Columns 43 through 49 
    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000    3.0000
  Column 50 
    3.0000

> format short e
We wee that by the 50th step, the error is 1.8e-9. 
> lambda-3
ans =
  Columns 1 through 6 
  3.0773e+000  1.0127e+000  5.0475e-001  2.8804e-001  1.7520e-001  1.1036e-001
  Columns 7 through 12 
  7.0961e-002  4.6214e-002  3.0342e-002  2.0026e-002  1.3262e-002  8.8023e-003
  Columns 13 through 18 
  5.8511e-003  3.8931e-003  2.5920e-003  1.7265e-003  1.1504e-003  7.6661e-004
  Columns 19 through 24 
  5.1095e-004  3.4057e-004  2.2702e-004  1.5134e-004  1.0089e-004  6.7255e-005
  Columns 25 through 30 
  4.4836e-005  2.9890e-005  1.9927e-005  1.3284e-005  8.8561e-006  5.9041e-006
  Columns 31 through 36 
  3.9360e-006  2.6240e-006  1.7493e-006  1.1662e-006  7.7749e-007  5.1832e-007
  Columns 37 through 42 
  3.4555e-007  2.3037e-007  1.5358e-007  1.0239e-007  6.8257e-008  4.5504e-008
  Columns 43 through 48 
  3.0336e-008  2.0224e-008  1.3483e-008  8.9885e-009  5.9924e-009  3.9949e-009
  Columns 49 through 50 
  2.6633e-009  1.7755e-009

Now try a matrix with a smaller gap between the largest and the
second largest eigenvalues. 
> A = [3,0,0;0,2.5,0;0,0,2]
A =
  3.0000e+000            0            0
            0  2.5000e+000            0
            0            0  2.0000e+000
> A = inv(B)*A*B
> A =
  3.5767e+000  7.6024e-001  7.6304e-001
 -2.6005e-001  2.3207e+000 -1.7925e-001
 -9.2359e-001 -8.5918e-001  1.6026e+000
> z = rand(3,1);
> for i=1:50, w = A*z; lambda(i) = w(1)/z(1); z = w/sqrt(sum(w.*w)); end
After 50 iterations, the error is larger... 5.2e-6 
> lambda-3
ans =
  Columns 1 through 6 
  2.3500e+000  8.9128e-001  4.6630e-001  2.7497e-001  1.7244e-001  1.1227e-001
  Columns 7 through 12 
  7.5010e-002  5.1113e-002  3.5411e-002  2.4901e-002  1.7762e-002  1.2846e-002
  Columns 13 through 18 
  9.4195e-003  7.0009e-003  5.2724e-003  4.0211e-003  3.1036e-003  2.4219e-003
  Columns 19 through 24 
  1.9089e-003  1.5180e-003  1.2165e-003  9.8146e-004  7.9637e-004  6.4931e-004
  Columns 25 through 30 
  5.3154e-004  4.3658e-004  3.5957e-004  2.9681e-004  2.4546e-004  2.0329e-004
  Columns 31 through 36 
  1.6857e-004  1.3991e-004  1.1622e-004  9.6601e-005  8.0335e-005  6.6835e-005
  Columns 37 through 42 
  5.5622e-005  4.6302e-005  3.8552e-005  3.2105e-005  2.6740e-005  2.2273e-005
  Columns 43 through 48 
  1.8555e-005  1.5458e-005  1.2879e-005  1.0730e-005  8.9405e-006  7.4496e-006
  Columns 49 through 50 
  6.2074e-006  5.1725e-006

take the gap smaller still... 
> A = [3,0,0;0,2.9,0;0,0,2]
A =
  3.0000e+000            0            0
            0  2.9000e+000            0
            0            0  2.0000e+000
> A = inv(B)*A*B
A =
  3.5657e+000  7.1758e-001  7.6215e-001
 -1.5153e-001  2.7392e+000 -1.7056e-001
 -1.0161e+000 -1.2159e+000  1.5952e+000
> z = rand(3,1);
> for i=1:50, w = A*z; lambda(i) = w(1)/z(1); z = w/sqrt(sum(w.*w)); end
After 50 iterations, the error is larger still... 9.2e-4 
> lambda-3
ans =
  Columns 1 through 6 
  2.7037e+000  9.5100e-001  4.8356e-001  2.7942e-001  1.7200e-001  1.0990e-001
  Columns 7 through 12 
  7.2034e-002  4.8177e-002  3.2828e-002  2.2812e-002  1.6210e-002  1.1823e-002
  Columns 13 through 18 
  8.8849e-003  6.9024e-003  5.5522e-003  4.6220e-003  3.9717e-003  3.5084e-003
  Columns 19 through 24 
  3.1706e-003  2.9172e-003  2.7210e-003  2.5638e-003  2.4335e-003  2.3221e-003
  Columns 25 through 30 
  2.2240e-003  2.1356e-003  2.0546e-003  1.9791e-003  1.9082e-003  1.8409e-003
  Columns 31 through 36 
  1.7768e-003  1.7154e-003  1.6566e-003  1.6000e-003  1.5455e-003  1.4930e-003
  Columns 37 through 42 
  1.4424e-003  1.3935e-003  1.3464e-003  1.3009e-003  1.2569e-003  1.2145e-003
  Columns 43 through 48 
  1.1735e-003  1.1340e-003  1.0958e-003  1.0588e-003  1.0232e-003  9.8873e-004
  Columns 49 through 50 
  9.5546e-004  9.2331e-004