How can I use the Solve command to find an eigenvector corresponding to a specific eigenvalue?

I have the following matrix in Mathematica:

L={{0, 0, (111/190), (79/190)},
{0.16, 0, 0, 0},
{0, 0.12, 0, 0},
{0, 0, 0.19, 0}}

Then using Eigenvalues[Transpose[L]], I'm able to get the eigenvalues of the transpose of L.

But I'm having a difficult time trying to solve for the eigenvector associated with the eigenvalue, 0.257651, using the solve command.

If you have an accurate estimate of an eigenvalue, you can find the corresponding eigenvector with NullSpace

NullSpace[L - (0.25765082710282156 + 0. I) IdentityMatrix]
(* {{0.812504 + 0. I, 0.504561 + 0. I, 0.234998 + 0. I, 0.173295 + 0. I}} *)

Eigensystem gives you eigenvalues and corresponding eigenvectors, no need for Solve:

Eigensystem[Transpose[L]]

{{0.257651, -0.0698441 + 0.212197 I, -0.0698441 - 0.212197 I, -0.117963},
{{-0.234715 + 0. I, -0.377966 + 0. I, -0.811528 + 0. I, -0.378777 + 0. I},
{-0.221686 + 0.163666 I, -0.120288 - 0.365452 I, 0.716245 + 0. I, 0.418347 + 0.296685 I},
{-0.221686 - 0.163666 I, -0.120288 + 0.365452 I, 0.716245 + 0. I, 0.418347 - 0.296685 I},
{-0.262679 + 0. I, 0.193664 + 0. I, -0.190376 + 0. I, 0.925879 + 0. I}}}
• Since I'm solving for a left eigenvector, would I have to take the transpose of {-0.234715 + 0. I, -0.377966 + 0. I, -0.811528 + 0. I, -0.378777 + 0. I}, for example? – K.M Apr 17 at 16:51
• No, the vectors in the second list are the left eigenvectors. Just try it out if you're unsure. And do read the documentation please. – Roman Apr 17 at 16:56
• The documentation on the Wolfram site? – K.M Apr 17 at 16:58