How to get from the the matrix to the constant?

Question

A is a symmetric matrix such that $ Av=\lambda v$.

I am not sure how to get from the first highlighted part to the second. The first one is a matrix whereas the second one is a number. At first I thought that A was replaced with $\lambda $ but I don't think this is possible because it would mean we are replacing a matrix with a constant.

Any help would be much appreciated!

Answer 1

We use the property that $v_\alpha$ is an eigenvector.

\begin{align} \left( A - \frac{\gamma}{\beta}I \right) \langle d \rangle &=\left( A - \frac{\gamma}{\beta}I \right)\sum_{\alpha=1}^N v_\alpha c_\alpha (t)\\ &=\sum_{\alpha=1}^N\left( Av_\alpha - \frac{\gamma}{\beta}v_\alpha \right) c_\alpha (t)\\ &=\sum_{\alpha=1}^N\left( \lambda_\alpha v_\alpha - \frac{\gamma}{\beta}v_\alpha \right) c_\alpha (t)\\ &=\sum_{\alpha=1}^N\left( \lambda_\alpha - \frac{\gamma}{\beta} \right)v_\alpha c_\alpha (t)\\ \end{align}