Revision as of 11:57, 11 September 2013 edit Slawekb (talk \| contribs) Extended confirmed users 11,467 edits →Decomposition: add back in the case of second order tensors, as this is the most often seen application of this decomposition. Rearranged material about general decompositions slightly. ← Previous edit		Revision as of 11:58, 11 September 2013 edit undo Slawekb (talk \| contribs) Extended confirmed users 11,467 edits m →Decomposition: using ''r'' in both places. Next edit →
Line 53: ==Decomposition== In full 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 is an integer ''nr'' and non-zero vectors ''v''<sub>1</sub>,...,''v''<sub>''nr''</sub> ∈ ''V'' such that :<math>T = \sum_{i=1}^nr \pm v_i\otimes v_i.</math> This is [[Sylvester's law of inertia]]. The minimum number ''nr'' for which such a decomposition is possible is the 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 [[ellipsoid]] representing the moment of inertia. Ellipsoids are examples of [[algebraic varieties]]; and so, for general rank, symmetric tensors, in the guise of [[homogeneous polynomial]]s, are used to define [[projective varieties]], and are often studied as such.

Symmetric tensor: Difference between revisions