Does this question make any sense - eigenvalues and norms

Question

Im having difficulties understanding this question:

show that if $b$ is an eigenvector of an invertible matrix $A$ with an eigenvalue $\lambda_1$ and $\delta b$ is an eigenvector of $A$ with an eigenvalue $\lambda_2$ then:

$$\frac{||\delta x||}{||x||}=\frac{|\lambda_1|* ||\delta b||}{|\lambda_2|* ||b||}$$

does this question make any sense? Whats $x$? is it just any vector? and what is $||.||$? some random norm?

Answer 1

If $x=A^{-1}b$, $\delta x=A^{-1}\delta b$, $Ab=\lambda_1b$, and $A\delta b=\lambda_2b$ (with $b\neq 0$ and $\delta b\neq 0$), then $x=b/\lambda_1$ and $\delta x=\delta b/\lambda_2$ and for any vector norm $\|\cdot\|$ we have $\|x\|=\|b\|/|\lambda_1|$ and $\|\delta x\|=\|\delta b\|/|\lambda_2|$.