tensor-decomposition

tensorslinear-algebramathematicsdecomposition

Abstraction: Introduction to tensors and their decomposition using Young diagrams

Key points:

A vector is a rank-1 tensor (one index), a matrix is a rank-2 tensor (two indices); a rank-3 tensor has 27 elements for n=3 and can be represented as three stacked matrices (flattening/matricizing).
The inner product of two rank-k tensors generalizes the vector dot product using a symmetric positive-definite metric matrix G.
Matrices decompose into symmetric and antisymmetric parts; analogously, rank-3 tensors decompose into subspaces parametrized by Young diagrams (fully symmetric, fully antisymmetric, and mixed-symmetry types).
Young diagram notation encodes symmetry type: boxes arranged so rows and columns are non-increasing; three distinct diagrams exist for rank-3 tensors.
Higher-rank tensor decompositions generalize this via representation theory of the symmetric group; references include Kolda & Bader (SIAM Review 2009) and Watkins (Matrix Computations).
Symmetrizing/antisymmetrizing operators T are projection operators: T(T(X)) = T(X) and T_sym composed with T_anti = 0.

Connections: Tensors · Tensor Decomposition · Linear Algebra