Confession:
After years of using them, I still don't really get what an eigenvalues and eigenvectors are conceptually. Something something special number something something basis vector 🤷

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Glad I’m not the only one lol

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They allow a square matrix A to be factored as A = UDU' where U is a matrix of eigenvalues and D is a diagonal matrix of eigenvalues. The cols of U have special othrormality properties. D may have special properties too depending on if A is symmetric or positive definite

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Typos didn’t help. U should be eigenvectors, not both eigenvalues. Also “orthonormality”, though still a mouthful and maybe not clear enough

Just finished a linear algebra course at my local community college, I know how they work in theory but OP how do you actually use them on a day to day basis?

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