Can someone provide an intuitive/conceptual explanation for eigenvalues and eigenvectors? I know the mathematical definition $$Ax= \lambda x$$ which means that if x is an eigenvector, then it is only stretched or squeezed (i.e., does not change direction). I feel like I don't really understand why that is so important. I am in an optimization class, and eigenvalues/eigenvectors come up a lot. For example, in quadratic programming we have the form $$x^T H x + c^T x \text{ subject to }Ax\leq b$$ and if H is a positive semidefinite matrix (non-negative eigenvalues), this problem is convex and easier to solve than the non-convex case. Why? I think I also read once that if there are negative eigenvalues, the matrix is "ill-conditioned." Is this true? What does that mean?
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1I sympathize with your question. I only truly learned the importance of eigenstuff after many years of study. Think of it this way: computing $Ax$ takes a lot of calculation. Add up how many multiplications and additions it takes to calculuate $Ax$ when everything is $n$-dimensional. Now do the same thing for computing $\lambda x$: it only takes $n$ multiplications, that's it. In some sense it is about efficiency. – Randall Sep 30 '22 at 14:40
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In many applications, eigenvalues and eigenvectors reveal some physical quantities giving you a lot of information about the stability of a system, convergence of an algorithm, etc.. – yes Sep 30 '22 at 14:41
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https://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors – Manish Saini Sep 30 '22 at 15:10
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https://math.stackexchange.com/q/243533/305862 – Jean Marie Sep 30 '22 at 16:55