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a tensor, say a 3X3, has nine components. Upon change of coordinate system, those 9 numbers change accordingly. If the change is a special rotation, then the nondiagonal elements vanish and only the diagonal ones remain. (diagonalization).

That sounds all good, but then I think about the stress tensor. The non-diagonal elements represent shear forces, which are physically real. Diagonalizing makes them disappear.

For a surface subject to both compressive and shearing forces, after diagonalization, we only see the compressive (or stretching) forces.

what am i missing? it seems that simplifying the mathematics through diagonalization hides the physical reality. I know it cannot be, but..

thank!

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# Components of a tensor

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