A Motivation Question

Why do all the arithmetic of calculating eigencomponents, when there is software available to do it for us  

The reason for this tedious working through is that the separations we have made by symmetry and by block structure do enable us to pick out and observe patterns in the eigenvectors which could be confused when there are two or more eigencomponents with the same eigenvalue. These patterns are going to be significant in a couple of chapters time, and it is important to observe them empirically first. 

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