Math REU Apprentice Program Alternate Proof of the Spectral Theorem
نویسنده
چکیده
This paper is a proof of the Spectral Theorem; a theorem that was proven in the apprentice class but not in this manner. Because the Spectral Theorem is such a fundamental theorem in linear algebra, it is important to explore it further through an alternate proof.
منابع مشابه
Chicago Math REU Apprentice Program Exercises Friday , July 24 Instructor : László Babai Scribe : Angela
Exercise 25.2. Recall that for A ∈Mn(R) with A = A , the Rayleigh quotient is defined as RA(x) = xTAx xT x . For symmetric φ : V → V , the Rayleigh quotient is defined as Rφ(x) = 〈x,φx〉 ||x||2 . Use the above theorem to show that argmaxxRφ(x) is achieved on V \ {0}. Proof. Note that Rφ(αx) = Rφ(x). So any value achieved by Rφ on V \ {0} is achieved on the unit sphere {x ∈ V | ||x|| = 1}. Restri...
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