The Spectral Theorem

The Complex Spectral Theorem and the Real Spectral Theorem, with examples.

[miro.s]

There is a missing case of equal eigenvalues and showing that we can still find perpendicular eigenvectors. It is trivial if we consider they lie in orthogonal eigenspace to other eigenvectors with different eigenvalues.

[nathanbonin]

Hi, this is really well presented but i feel like the difficulty is hidden in shur's theorem at 5:30, it would be interessting to develop this point. Thank you for your work

[nohaal-mahrooqi]

If the vector space is Complex then the orthonormal basis is the same as eigenvector of the Matrix?

[jessefranckowiak]

There's a typo at the 1:50 mark: "For which operators on V is there *is* an orthonormal basis basis of V"

[kgeorge]

4:24 so if the matrices of any two operators with respect to some basis commute, then the operators itself commute as well? (tried to prove this, seems reasonable, but not at all obvious)

[garfieldnate]

I was following the proof of theorem 7.27 until I got to the words, "Thus the equation above implies that m > 0..." I can follow before and after this point, but I don't understand what implies that m > 0.

[edwardhartz]

I wish he would reply to comments. They are good videos but I haven’t seen him answering anyone’s questions.