Mathematica only needs one function to get all eigenvalues and corresponding eigenvectors:
Intrinsic systems [{{ 1, 1/3, 1, 1/6, 1/5, 1/3, 1/2},
{3, 1, 3, 1/3, 1/4, 1/2, 1/2},
{ 1, 1/3, 1, 1/5, 1/5, 1/5, 1/3},
{6, 3, 5, 1, 1, 2, 1},
{5, 4, 5, 1, 1, 2, 2},
{3, 2, 5, 1/2, 1/2, 1, 1/2},
{2, 2, 3, 1, 1/2, 2, 1}}]
The numerical results are given by the following formula:
N[ Intrinsic system [{{ 1, 1/3, 1/6, 1/5, 1/3, 1/2},
{3, 1, 3, 1/3, 1/4, 1/2, 1/2},
{ 1, 1/3, 1, 1/5, 1/5, 1/5, 1/3},
{6, 3, 5, 1, 1, 2, 1},
{5, 4, 5, 1, 1, 2, 2},
{3, 2, 5, 1/2, 1/2, 1, 1/2},
{2, 2, 3, 1, 1/2, 2, 1}}]]
The output is
{{7.24398，-0.0507673+ 1. 19 136 I，-0.0507673- 1. 19 136 I，-0.00 12 1045+0.444738 I，-0.0065438
{{0.28 126, 0.539823, 0.249504, 1.40788, 1.58 162, 0.827308, 1.},
{-0.044354 - 0.25 1286 I，-0.394803 + 0. 1986 1 I，-0.093876 1. 124568 I，0. 102755 + 0.484544 I，0.45656565438+.},
{-0.044354 + 0.25 1286 I，-0.394803 - 0. 1986 1 I，-0.093876 1. 124568 I，0. 102755 - 0.484544 I，0.45656565438+.},
{-0.0238 199 - 0.332635 I，0.0398586-0.5 1 1946 I，0.2 10527 + 0.4053 17 I，- 1.9436+ 1.83 159 I，-2},
{-0.0238 199 + 0.332635 I，0.0398586+0.5 1 1946 I，0.2 10527 - 0.4053 17 I，- 1.9436- 1.83 159 I，-2},
{-0.58720 1-0. 130567 I，0.4 1444 1.3409 1 1 I，0.4522 18 + 0.0228507 I，-0.80 1689 + 4.0565438},
{-0.58720 1+0. 130567 I，0.4 1444 1-0.3409 1 1 I，0.4522 18 - 0.0228507 I，-0.80 1689 - 4.0565438}}}
The first group is eigenvalues, followed by corresponding eigenvectors.
So there is only one real eigenvalue: 7.24398, and the corresponding eigenvector:
{0.28 126, 0.539823, 0.249504, 1.40788, 1.58 162, 0.827308, 1.}.
I just saw the same problem (but the positions of (1, 6) and (6, 1) are different).
Is characteristic vector multiplied by non-zero number or characteristic vector?
As a weight, do you want the sum of the components to be 1?
Then we might as well divide the eigenvector obtained above by the sum of the components.
{0.05 10222, 0.0927348, 0.0426905, 0.24047, 0.269532, 0. 134296, 0. 169255}.