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(In this question, $(*)$ means normal when working over $\mathbb C$ and means self-adjoint when working over $\mathbb R$.)

This question is related but despite the same title what that question actually asks is what is the importance of diagonalizing a matrix. I think we can all agree that diagonalization of matrices is useful. Is there anything more to the spectral theorem than knowing when a linear operator can be diagonalized with respect to some orthonormal basis?

To further motivate this question, here's a quote from the book Linear algebra done right:

The Spectral Theorem is probably the most useful tool in the study of operators on inner product spaces.

It certainly is useful for $(*)$ operators. But is it useful to study operators that aren't $(*)$? It would seem weird to call it the "most useful tool" in this area if it is only useful for this kind of operators, which are in a sense the "simplest" ones.

An answer to this question could either provide intuition for the importance of the Spectral Theorem or give examples of applications of it to non-$(*)$ operators. Answers (only) about the importance of diagonalizing matrices are obviously not welcome since we already agreed on the usefulness of that.

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  • $\begingroup$ No, the theorem doesn't apply in general to operators that are not (*). But note, for example, that if you start with a skew-symmetric real matrix $A$, then $iA$ will be hermitian and the spectral theorem will apply to it, yielding important results. $\endgroup$ Feb 21, 2023 at 17:00
  • $\begingroup$ Implicit in your quote is that (*) operators are the most useful operators. $\endgroup$ Feb 21, 2023 at 17:07

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The Spectral Theorem leads to the Singular Value Decomposition and to the Polar Decomposition, both of which are hugely important and apply to all operators, not just self-adjoint or normal operators.

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