TITLE: Intuition on spectral theorem
QUESTION [22 upvotes]: In the last month I studied the spectral theorems and I formally understood them. But I would like some intuition about them. If you didn’t know spectral theorems, how would you come up with the idea that symmetric/normal endomorphisms are the only orthogonally diagonalizable endomorphisms in the real/complex case. How would you even come up with the idea of studying the adjoint?

REPLY [16 votes]: Almost everything about this subject was derived in the opposite order of what you have been taught. That's why it is difficult to answer your question.

The infinite-dimensional case was studied for functions before the finite-dimensional case, and well before the notion of a vector space.

Orthogonality was noticed and defined using integral conditions about 150 years before an inner product was defined, and before finite-dimensional Linear Algebra. These observations led to the notion of a general inner product space.

Linearity came out of the physical condition of superposition of solutions for the Heat Equation and vibrating string problem, not the other way around.

Self-adjoint was defined before there was an inner-product, through Lagrange's adjoint equation, which gave, among other things, a reduction of order tool for ODEs, and a notion of "integral orthogonality."


It's all upside down from the point-of-view of abstraction. Asking how you might start at the lowest level of abstraction and naturally move toward the more abstract direction is asking how to motivate the backwards direction from the Historical forward direction that brought us to this point. It wasn't derive that way, and might never have been.