Examples
Many material properties and fields used in physics and engineering can be represented as symmetric tensor fields; for example, stress, strain, and anisotropic conductivity.
In full analogy with the theory of symmetric matrices, a (real) symmetric tensor of order 2 can be "diagonalized". More precisely, for any tensor T ∈ Sym2(V), there is an integer n and non-zero vectors v1,...,vn ∈ V such that
This is Sylvester's law of inertia. The minimum number n 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 polynomials, are used to define projective varieties, and are often studied as such.
Read more about this topic: Symmetric Tensor
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