In Quantum Mechanics, Ω is called an 𝐨𝐛𝐬𝐞𝐫𝐯𝐚𝐛𝐥𝐞 only if it can be represented by a Hermitian operator acting on the Hilbert space.pic.twitter.com/SLvTZ3SPUT
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Property 1: Hermitian operators have all real eigenvalues. Here Ω is a hermitian operator and |Ωᵢ〉is one of its eigenvectors.pic.twitter.com/Jk3NmPfdN0
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Property 2: Eigenvectors of a hermitian operator Ω with *distinct* eigenvalues are orthogonal.pic.twitter.com/fceXleLAB7
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Isn't this true for symmetric matrices? This allows you to do the spectral decomposition.
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When your field is the reals Hermitian = symmetric
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