Levi-Civita symbol

In three dimensions, ${ϵ}_{ijk}$ is zero when indices repeat and is the sign of the permutation $(i,j,k)$ otherwise.

For indices in $\{1,\dots ,n\}$, this Levi-Civita symbol is zero when any indices repeat, and otherwise is the sign of a permutation taking $(1,\dots ,n)$ to $(i_1,\dots ,i_n)$. Thus ${ϵ}_{1\cdots n}=1$ and exchanging any two index slots changes the sign. It expresses a determinant as $\det M={ϵ}_{i_1\cdots i_n}{M}_{i_1,1}\cdots M_{i_n,n}$ with repeated indices summed. Exactly $n!$ components are nonzero, so $\sum {ϵ}_{i_1\cdots i_n}^2=n!$.

The determinant transformation formula for the Levi-Civita symbol shows that it is invariant under every proper rotation matrix, for which $\det R=1$. Under an orthogonal reflection its sign reverses. This distinguishes proper-rotation isotropy from the pseudotensor transformation convention when both orientations are included.

Contracting one index gives

For a $3\times 3$ matrix $A$, contraction of ${ϵ}_{lmn}{A}_{il}{A}_{jm}{A}_{kn}=(\det A){ϵ}_{ijk}$ with ${ϵ}_{ijk}$ proves the title identity, because ${ϵ}_{ijk}{ϵ}_{ijk}=6$. DefineExpansion shows that $C$ is the cofactor matrix. Equivalently, the determinant identity gives $A_{ib}{C}_{ab}=(\det A){\delta }_{ia}$. Thus for invertible $A$,Differentiating the cubic determinant formula gives three equal terms, so $\partial (\det A)/\partial A_{ab}=C_{ab}$. This is the entrywise derivative of the determinant and remains valid for singular $A$ through the cofactor expression. The inverse expression requires $\det A\ne 0$.

For three-dimensional vectors,The corresponding identity in ${\mathbb{R}}^n$ is

For real three-dimensional vectors, Lagrange's identity gives $|a\times b{|}^2=a^{T}Ma$ with symmetric matrix $M=|b{|}^{2}I-b{b}^T$. For $b\ne 0$, the normal line spanned by $b$ is its kernel and the perpendicular plane is its $|b{|}^2$ eigenspace. Consequently $M$ is positive semidefinite, has rank two and has zero determinant. For $b=0$, it is the zero matrix.

For three vectors in oriented Euclidean three-space, the scalar triple product is $a\cdot (b\times c)$ and the vector triple product is $a\times (b\times c)$. The former is their signed volume; the latter is the vector operation appearing in the vector triple product identity.

Contracting two Levi-Civita symbols gives