Sunday, September 27, 2015

linear algebra - Maximum eigenvalue of a hollow symmetric matrix




Is the maximum eigenvalue (or spectral radius) of the matrix with the following form equalled to row or column sum of the matrix?




$$
A=\left( \begin{array}{cccc}
0 & a & ... & a \\
a & 0 & ... & a \\
: & : & ...& : \\
a & a & ... & 0\end{array} \right) $$



The matrix is square with dimension $n \times n$ where $n = 2,3,4,...$, hollow (all elements in the principal diagonal = 0), symmetric and all off diagonal elements have the same value.




Is the spectral radius of such matrices = $(n-1)\times a$? Why?


Answer



Start with the matrix $A$ of all $a$'s, whose eigenvalues are zero except for eigenvalue $na$ having multiplicity one (because rank$(A) = 1$).



Now subtract $aI$ from $A$ to get your matrix. The eigenvalues of $A-aI$ are those of $A$ shifted down by $a$. We get a eigenvalue $(n-1)a$ of multiplicity one and eigenvalue $-a$ with multiplicity $n-1$.



So the spectral radius (largest absolute value of an eigenvalue) of $A$ is $|na|$, and the spectral radius of $A-aI$ is $\max(|(n-1)a|,|a|)$. The latter is simply $|(n-1)a|$ unless $n=1$.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...