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But I don't really understand "what they are". How is a matrix with complex eigenvalues any different than a matrix with real eigenvalues? What does the eigenvalue tell us about the form of a matrix? What does it tell us about its character - its form?

I've always understood things like derivatives, because they make sense. The derivative of a function is just how steep its slope is. But linear algebra just doesn't make sense to me. What is an eigenvalue? What is a determinant? What is the matrix?