Question

2. The spectral decomposition theorem states that the eigenstates of any Hermitian matrix form an orthonormal basis for the linear space. Let us consider a real 3D space where a vector is denoted by a 3x1 column vector. Consider the symmetric matrix B-1 1 1 Show that the vectors 1,0, and1are eigenvectors of B, and find 0 their eigenvalues. Notice that these vectors are not orthogonal. (Of course they are not normalized but let's don't worry about it. You can normalize them if you insist but without doing so will save you some writing.) Explain why this does not contradict the theorem which states that eigenvectors of a Hermitian (or symmetric) matrix are orthogonal if they correspond to distinct eigenvalues. Note also that this set does not agree with the spectral decomposition theorem. Knowing that what mathematicians told us must be correct, how can you fix this discrepancy, i.e., how can you construct an orthogonal set of eigenvectors of B from the above three vectors?

Answer 1

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