Find maximum and minimum eigenvalues of symmetric matrix

Question

I have to find maximum and minimum eigenvalues of symmetric matrix. First that I've done is tridiagonalize it and now it looks like $$ {A_{n \times n}} = \left[ {\begin{array}{*{20}{c}} {a_1}&{b_2}&{}&{}&{}\\ {b_2}&{a_2}&{b_3}&{}&{}\\ {}&{b_3}&{\ddots}&{\ddots}&{}\\ {}&{}&{\ddots}&{\ddots}&{b_{n}}\\ {}&{}&{}&{b_{n}}&{a_n} \end{array}} \right] $$ I have to do it with some kind of recursion but can't find anything that can help me to understand how to do it.

Answer 1

The key is that the task is formulated not properly. I thought I should use some properties of tridiagonal matrices but I don't. To find max and min eigenvalues it's enough to use simple iterations: $$ x^{(0)} = (x_1, x_2, \ldots, x_n)^T $$ $x^{(0)}$ is a random vector. $$ x^{(k+1)} = A \frac{x^{(k)}}{\alpha_k} $$ Where $\alpha_k$ is the maximum element of $x^{(k)}$ in absolute value. So, $\lambda_{max}=\alpha_n$ is the max eigenvalue in absolute value.

To find the minimum eigenvalue: $$ x^{(k+1)} = A^{-1} \frac{x^{(k)}}{\alpha_k} $$ Where $\alpha_k$ is the maximum element of $x^{(k)}$ in absolute value. So, $\lambda_{min}=\frac{1}{\alpha_n}$ is the min eigenvalue in absolute value.