Multiinear Algebra

d-tensor

a : F_{n_{1}, ⋯, n_{d}} .

a_{i₁, ⋯, i_d} for i_j = 1 ⋯ n_j for j = 1 ⋯ d.

A scalar is a 0-tensor. A vector is a 1-tensor. A matrix is a 2-tensor.

[Lecture 1]

\begin{array}{l} f : & F_{n_{1}, n_{2}, n_{3}} & → F, \\ a : & F_{n_{1}, n_{2}, n_{3}} & . \end{array}

f (x_{1}, x_{2}, x_{3}) = ...

D_{x} (f) ≡ (a_{i_{1}} x_{3} ⊗ x_{2}, a_{i_{2}} x_{3} ⊗ x_{1}, a_{i_{3}} x_{2} ⊗ x_{1}) .

[Lecture 2]

Multilinear Map

\begin{array}{l} A_{1} & , ⋯, & A_{n} & ,  B ⊆ & L_{F} \\ f : & A_{1} & × ⋯ × & A_{n} & → B. \end{array}

Tensor

Tensor Product

\begin{matrix} A, B ⊆ L & ( & f & ) & , \\ A ⊗ B \end{matrix}

Tensor Field

Alternating Multilinear Map

a ∧ b = (k + m)! {(k! m!)}^{−1} alt (a ⊗ b),

alt (a) (x_{1}, ⋯, x_{k}) = {(k!)}^{−1} ε ....

Interior Product

Tensor Contraction

\begin{array}{l} A : & {L (F)}_{n} & , & A^{*} & : A → F, \\ C : & V^{*} & ⊗ V → F. \end{array}

Symmetric Tensor

a_{i_{1}, ⋯, i_{d}} = a_{i_{p (1)}, ⋯, i_{p (d)}} .

Antisymmetric Tensor

a_{i_{1}, ⋯, i_{d}} = − a_{i_{p (1)}, ⋯, i_{p (d)}} .

Penrose Graphical Notation

Tensor Decomposition

Tensor Algebra

T^{k} L = L^{⊗ k} = L ⊗ ⋯ ⊗ L.

T (L) = ⊕_{k} T^{k} L = K ⊕ L ⊕ (L ⊗ L) ⊕ ⋯.

Mixed Tensor

The covariant metric tensor, contracted with a (m, n)-tensor, yields a (m − 1, n + 1)-tensor, whereas its contravariant inverse, contracted with a (m, n)-tensor, yields a (m + 1, n − 1)-tensor.

{T_{i, j}}^{k} = T_{i, j, l} g^{l, k} .

{T_{i, j}}^{k} Δ^{k, l} = {T_{i, j}}^{l} .

Multivector

Multilinear Form

References

From Matrix To Tensor: The Transition to Computational Multilinear Algebra. Charlie Van Loan. 2010.