Symmetric tensor: Difference between revisions

Revision as of 11:58, 11 September 2013

Revision as of 12:04, 11 September 2013

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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'' ∈ Sym²(''V''), there is an integer ''r'' and non-zero vectors ''v''₁,...,''v''_''r'' ∈ ''V'' such that

:<math>T = \sum_{i=1}^r \pm v_i\otimes v_i.</math>

This is [[Sylvester's law of inertia]]. The minimum number ''r'' 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 [[Poinsot's ellipsoid]] representing the moment of inertia.

Ellipsoids[[Ellipsoid]]s 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.

For symmetric tensors of arbitrary order ''k'', decompositions