I'm currently taking a conceptual course in linear algebra, and I'm trying to understand why it would be theoretically useful or illuminating to know when a linear operator (or its matrix representation) is diagonalizable. Why is this? How does it help us to more deeply understand vector spaces, linear operators, their applications, etc.?
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3Possible duplicate: What is the motivation defining Matrix Similarity?. – Git Gud Feb 06 '15 at 19:27
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4Diagonalizable matrices are very convenient to work with computationally. If you can write $A = P D P^{-1}$, the powers of $A$ can be evaluated very quickly. – Umberto P. Feb 06 '15 at 19:31
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See also https://math.stackexchange.com/questions/23312/what-is-the-importance-of-eigenvalues-eigenvectors – lhf Jan 08 '19 at 13:46