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→Decomposition: Edited back-in the order 2 decomposition (it is probably indeed good to state it explicitly), but made it slightly more general and more in line with the general case. |
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==Decomposition==
In analogy with the theory of [[symmetric matrix|symmetric matrices]], a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor ''T'' ∈ Sym<sup>2</sup>(''V''), there are an integer ''r'', non-zero unit vectors ''v''<sub>1</sub>,...,''v''<sub>''r''</sub> ∈ ''V'' and weights ''λ''<sub>1</sub>,...,''λ''<sub>''r''</sub> such that
:<math>T = \sum_{i=1}^r \lambda_i \, v_i\otimes v_i.</math>
The minimum number ''r'' for which such a decomposition is possible is the (symmetric) rank of ''T''. The vectors appearing in this minimal expression are the ''[[principal axes]]'' of the tensor, and generally have an important physical meaning. For example, the principal axes of the [[inertia tensor]] define the [[Poinsot's ellipsoid]] representing the moment of inertia. Also see [[Sylvester's law of inertia]].
For symmetric tensors of arbitrary order ''k'', decompositions
:<math>T = \sum_{i=1}^r \lambda_i \, v_i^{\otimes k}</math>
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