Characteristic polynomial of an $n \times n$ matrix with only one column and one row non zero.

characteristic polynomialdeterminanteigenvalues-eigenvectorslinear algebraproblem solving

The problem asks for the eigenvalues of the matrix

$$\left(\begin{array}{cccc}
0 & \cdots & 0 & 1 \\
\vdots & & \vdots & \vdots \\
0 & \cdots & 0 & n-1 \\
1 & \cdots & n-1 & n
\end{array}\right).$$

Since it was an $n \times n$ matrix, and I can't think of something immediatly that says this is a diagonalisable matrix so that I can use the trace and the product of the diagonal elements to deduce the eigen values, I thought about applying the general rule $\operatorname{det}(A)=\sum_{\sigma \in S_{n}}\left(\operatorname{sgn}(\sigma) \prod_{i=1}^{n} a_{i, \sigma_{i}}\right)$

So it was something along: $ -\lambda \times \det\left(\begin{array}{cccc}
-\lambda & \cdots & 0 & 2 \\
\vdots & \ddots & \vdots & \vdots \\
0 & \cdots & -\lambda & n-1 \\
2 & \cdots & n-1 & n – \lambda
\end{array}\right) \pm \det\left(\begin{array}{cccc}
0 & -\lambda & 0 & \cdots \\
0 & & \ddots & 0 \\
\vdots & & 0 & -\lambda \\
1 & \cdots & 0 & n-1 \\
\end{array}\right) $

I didn't continue along this it just didn't look like the best or practical way to approach it.
Are there any particular properties I can use to shortcut that I couldn't see here?

The solution to the problem is (which moreover looked a bit far from where I was heading, although I don't see where the 6 is coming from and how it makes sense) :

$ 0 \text { and the roots of } 6 \lambda^{2}-6 n \lambda-n(n-1)(2 n-1)=0 \text {. }$

Best Answer

Let's call your matrix $A$. The matrix $A$ is real and symmetric and so it is similar to diagonal matrix with entries $\lambda_1, \dots, \lambda_n$. Also, $A$ has rank two so the diagonal matrix must also have rank two so let's say $\lambda_3 = \dots = \lambda_n = 0$. Now you need two equations to find $\lambda_1,\lambda_2$. A possible set of such equations is:

  1. $\operatorname{tr}(A) = \lambda_1 + \lambda_2 + \dots + \lambda_n = \lambda_1 + \lambda_2 = n$.
  2. $\operatorname{tr}(A^2) = \lambda_1^2 + \lambda_2^2 + \dots + \lambda_n^2 = \lambda_1^2 + \lambda_2^2.$

One readily computes $\operatorname{tr}(A^2)$ to be $$ \operatorname{tr}(A^2) = 1^2 + 2^2 + \dots + (n-1)^2 + 1^2 + 2^2 + \dots + n^2 = \frac{n(2n^2+1)}{3}. $$ By plugging $\lambda_2 = n - \lambda_1$ into $\lambda_1^2 + \lambda_2^2 = \operatorname{tr}(A^2)$ you get a quadratic equation for $\lambda_1$ whose solutions are $$ \frac{n}{2} \pm \sqrt{\frac{8n^3 - 6n^2 + 4n}{12}} $$ and those are the eigenvalues.

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