# 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.

## 2 Answers

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