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?
2026-03-26 04:50:42.1774500642
Conceptual meaning of eigenvectors and eigenvalues
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