Strange numerical values of Eigenvectors

Question

When I calculate Eigenvalues and Eigenvector by using:

Eigensystem[{{3.8, 21, 21}, {0.3, 3.5, 2.5}, {-0.8, -6.2, -5.2}}]

I get the following result:

{{1., 0.8, 0.3}, {{5.4769*10^-14, -0.707107, 0.707107}, {0.889001, 0.254, -0.381}, {-0.937043, -0.156174, 0.312348}}}

The outputs for the eigenvectors are strange.

The result should be:

Eigenvalues: 1, 3/10, 4/5

Eigenvector: {0,-1,1}, {-3, -0.5, 1}, {-7,-2,3}

My question is: how to do to get

{0,-1,1}, {-3, -0.5, 1}, {-7,-2,3} 

instead of

{5.4769*10^-14, -0.707107, 0.707107}, {0.889001, 0.254, -0.381}, {-0.937043, -0.156174, 0.312348} ?

I tried to convert the result by using Round, Clip, Accuracy, Floor, IntegerPart, N, Precision, SetPrecision with no success.

Thanks.

Answer 1

As stated in the Details section of the Eigensystem and Eigenvectors help page, the eigenvectors will be normalized to 1 for approximate numerical matrices. Nubers like 3.5 are approximate numbers. Replace your numbers with exact numbers and it should give you more visually pleasing eigenvectors in most cases.

Example:

Eigensystem[{{3 + 8/10, 21, 21}, {3/10, 3 + 5/10, 2 + 5/10}, {-8/10, -(6 + 2/10), -(5 + 2/10)}}]

Gives:

{{1,4/5,3/10},{{0,-1,1},{-7/3,-2/3,1},{-3,-1/2,1}}}