# Multiinear Algebra

d-tensor

$a : Fn1, ⋯, nd.$

ai1, ⋯, id for ij = 1 ⋯ nj for j = 1 ⋯ d.

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

[Lecture 1]

$f : Fn1, n2, n3 → F,a : Fn1, n2, n3.$$f(x1, x2, x3) = ...$$Dx(f) ≡ (ai1 x3 ⊗ x2, ai2 x3 ⊗ x1, ai3 x2 ⊗ x1).$

[Lecture 2]

## Multilinear Map

$A1, ⋯, An, B ⊆ LFf : A1 × ⋯ × An → B.$

## Tensor Product

$A, B ⊆ L(f),A ⊗ B$

## Alternating Multilinear Map

$a ∧ b = (k + m)! (k! m!)−1 alt(a ⊗ b),$$alt(a)(x1, ⋯, xk) = (k!)−1 ε ....$

## Tensor Contraction

$A : L(F)n, A* : A → F,C : V* ⊗ V → F.$

## Symmetric Tensor

$ai1, ⋯, id = aip(1), ⋯, ip(d).$

## Antisymmetric Tensor

$ai1, ⋯, id = −aip(1), ⋯, ip(d).$

## Tensor Algebra

$Tk L = L⊗ k = L ⊗ ⋯ ⊗ L.$$T(L) = ⊕k Tk 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.

$Ti, jk = Ti, j, l gl, k.$$Ti, jk Δk, l = Ti, jl.$