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Let $b$ be an eigenvector of $A$ with eigenvalue $\lambda_1$ and let $\delta b$ be an eigenvector of $A$ with eigenvalue $\lambda_2$. $A$ is invertible. Show that ${||\delta x||\over ||x||}={|\lambda_1|\over |\lambda_2|}\cdot {||\delta b||\over ||b||}$. I guess $\delta$ is supposed to be some matrix and that the norm is $l_2$. What I entirely didn't understand is the solution, as follows:

"If $\lambda$ is an eigenvalue of $A$ then ${1\over \lambda }$ is an eigenvalue of $A^{-1}$. Therefore, ${||\delta x||\over ||x||}=$ ${||A^{-1}\delta b||\over ||A^{-1}b||}$ $={|\lambda_1|\over |\lambda_2|}{||\delta b||\over ||b||}$. I understand all but the second first equality. I have no idea how one replaces $x$ by $b$. Could you help me figure it out?

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    your question raises an important question about this problem: what are $\delta x$ and $x$? – 2017-02-21
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    Vectors, is this the direction? – 2017-02-21
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    It literally came out as addressing a question to a vector... – 2017-02-21
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    Since you don't know what $\delta x$ and $x$ are, perhaps you can tell us where this problem came from? – 2017-02-22
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    It actually wasn't clarified, neither in the question nor the answer. I still look any reference up in the notes – 2017-02-22

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No, $δ$ and $\Delta$ in general indicate a difference. The general setting is that if $Ax=b$ has a unique solution for all $b$ and you compare to the solution of $A\tilde x=b+δb$ where $δb$ is assumed to be a small perturbation, then $\tilde x=x+δx$ with $δx$ also small and you want to express the relative error of $x$ in terms of the relative perturbation of $b$.

In the given situation it is used/demonstrated that this gives expressions in terms of (invariant) properties of $A$ if $b$ and $δb$ are eigenvectors of $A$. As the domain and range vector spaces need not be identical, they only are required to have identical dimension, the eigenvalues are not really an invariant property. It would be better to use right singular vectors so that the quotient is of singular values.

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    The notation in the notes I had was different, and no clarification was provided in the exercise. You just saved me a relatively enormous amount of time. Thank you so much! – 2017-02-22