I am looking at some notes on Linear algebra written for maths students mainly to improve my Quantum Mechanics. I came across the following example - $$ \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} $$(adsbygoogle = window.adsbygoogle || []).push({});

The example then gives the eigenvalues as 0 and 1(doubly degenerate). It then calculates the eigenvectors using Gaussian elimination. This is where my problem arises - coming from a physics background I tried to find the eigenvectors for the repeated eigenvalue 1 using back substitution but it doesn't seem to produce a solution this way. Am I doing something wrong or is it possible for back substitution not to work while Gaussian elimination works ?

The answer given for the eigenvector is a linear combination of the 2 vectors ( 3 1 0 )^{T}and (-1 0 1)^{T}. In the Quantum Mechanics textbook I am using it says for degenerate eigenvalues to choose 2 mutually orthogonal vectors. The 2 vectors I have listed are not orthogonal. Is the orthogonal part just a preference for QM and not a requirement ?

Thanks

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# I Eigenvectors for degenerate eigenvalues

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