# What is the second largest eigenvalue of Perron matrix.

$$\left[ \begin{array}{@{}ccccc@{}} 0.9& 0.1& 0& 0& 0& 0& \\ 0& 0.9& 0.1& 0& 0& 0& \\ 0& 0& 0.9& 0& 0& 0.1& \\ 0& 0& 0& 0.9& 0.1& 0& \\ 0& 0& 0& 0.1& 0.9& 0& \\ 0.1& 0& 0& 0& 0& 0.9& \\ \end{array} \right]$$

This is a Perron (P) Matrix, in documents i read μ_2 is the second largest eigenvalue of P matrix. here what is the μ_2 or second largest eigenvalue.

• I am sorry Jyrki I gave a wrong example i changed my matrix example.but i want ask an eigenvalue is a sum of a row or a column? – Toto Mar 25 '13 at 15:02
• No worries. Unfortunately withot the circulant structure it is hard to tell what the eigenvalues are :-/ – Jyrki Lahtonen Mar 25 '13 at 15:07

There is an excellent comment and response, but I'll write them out explicitly.

For the new matrix, you have:

$\lambda_1 = 1$

$\lambda_2 = 1$

$\lambda_3 = 0.9 + 0.1 i$

$\lambda_4 = 0.9 - 0.1 i$

$\lambda_5 = 0.8$

$\lambda_6 = 0.8$

What do you mean by largest eigenvalue in this context? Do you mean in absolute value, in magnitude or using some other measure?

• I mean absolute value but not largest ,largest 2nd eigenvalue..Could i ask how can you calculate λ1=1..λ6=0.8 ? Thank you so much – Toto Mar 25 '13 at 15:44
• The standard way $|A -\lambda I| = 0$ and then solve for the roots of the characteristic polynomial. Regards – Amzoti Mar 25 '13 at 15:52
• You are very welcome. Regards – Amzoti Mar 25 '13 at 17:06
• Amzoti I wonder do you have knowledge about primitive matrix,if yes is this matrix is primitive? – Toto Mar 25 '13 at 17:08
• Read the 1-st paragraph en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem. If still not clear, this can be a new question on how to determine it. Regards – Amzoti Mar 25 '13 at 17:19

This is a circulant matrix; the Wikipedia article gives the eigenvectors and eigenvalues.

• You are right it is my fault let me give a better example or edit the matrice.Thank you Joriki – Toto Mar 25 '13 at 14:49