# Levi-Civita symbol

In mathematics, a **Levi-Civita symbol** (or **permutation symbol**) is a quantity marked by *n* integer labels. The symbol itself can take on three values: 0, 1, and −1 depending on its labels. The symbol is called after the Italian mathematician Tullio Levi-Civita (1873–1941), who introduced it and made heavy use of it in his work on tensor calculus (*Absolute Differential Calculus*).

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## [edit] Definition

The Levi-Civita symbol is written as

The symbol designates zero if two or more indices (labels) are equal. If all indices are different, the set of indices forms a permutation of {1, 2, ..., *n*}. A permutation π has parity (signature): (−1)^{π} = ±1; the Levi-Civita symbol is equal to (−1)^{π} if all indices are different. Hence

else

**Example**

Take *n* = 3, then there are 3^{3} = 27 label combinations; of these only 3! = 6 give a non-vanishing result. Thus, for instance,

while

## [edit] Application

An important application of the Levi-Civita symbol is in the concise expression of a determinant of a square matrix.
Write the matrix **A** as follows:

then the determinant of **A** can be written as:

where Einstein's summation convention is used: a summation over a repeated upper and lower index is implied. (That is, there is an *n*-fold summation over *i*_{1}, *i*_{2}, ..., *i*_{n}).

## [edit] Properties

Very useful properties in the case *n* = 3 are the following,

Note that the sum in the first expression contains one non-zero term only: if *i* ≠ *j* there is one value left for *k* for which ε_{ijk} ≠ 0. The same holds for the second factor in the first expression. The sum over *k* is a convenient way of picking the value of *k* that gives a non-vanishing result. The double sum in the second expression is over two non-zero terms: ε_{ipq}ε_{jpq} and ε_{iqp}ε_{jqp}. The triple sum in the third expression is over 3!=6 non-zero terms.

### [edit] Proof

The proof of the properties is easiest by observing that ε_{ijk} can be written as a determinant. This also opens the way to a generalization for general *n* > 3.

Write

Obviously, the unit columns are orthonormal,

where δ_{ij} is the Kronecker delta.

Consider determinants consisting of three columns selected out of the three unit columns. Then by the properties of determinants:

Further,

Hence

Introduce 3×3 matrices **A** and **B** as short-hand notations:

Use

and

The zeros in the third column appear because *i* ≠ *k* and *j* ≠ *k*. (If this were not the case
ε_{ijk} = 0). A similar reason explains the zeros in the third row.
Hence,

A generalization of the property to arbitrary *n* is clear now:

The second property of the Levi-Civita symbol follows from

The determinant of the last matrix is equal to δ_{ij}. The same holds for *p* and *q* interchanged. In the case of general *n* the sum is over (*n*−1)! permutations [note that (3-1)!=2]. The final property contains a summation over six (3!) non-zero terms; each term is the determinant of the identity matrix, which is unity.

## [edit] Is the Levi-Civita symbol a tensor?

In the physicist's conception, a tensor is characterized by its behavior under transformations between bases of a certain underlying linear space. If the most general basis transformations are considered, the answer is *no*, the Levi-Civita symbol is *not* a tensor. If, however, the underlying space is proper Euclidean and only orthonormal bases are considered, then the answer is *yes*, the Levi-Civita symbol *is* a tensor.

In order to clarify the answer, it is necessary to consider how the Levi-Civita symbol behaves under basis transformations.

### [edit] Transformation properties

Consider an *n*-dimensional space *V* with non-degenerate inner product. Let two bases of this space be connected by the non-singular basis transformation **B**,

where by summation convention a sum over *i*_{k} is implied. The primes indicate a set of axes and may not be used for anything else. An arbitrary vector *a*∈*V* has the following components with respect to the two bases:

Consider a set of *n* linearly independent vectors with columns **a**_{k} and **a′**_{k} with respect to the unprimed and primed basis, respectively,

Take determinants,

Use

for *k* = 1, 2, ..., *n*, successively. Then

Since the component vectors **a′**_{k} are linearly independent, the coefficients of the powers of *a*^{i ′}_{i} may be equated and the following *transformation rule for the Levi-Civita symbol* results,

Except for the factor det(**B**), the symbol transforms as a covariant tensor under basis transformation. When only transformations with det(**B**) = 1 are considered, the symbol is a tensor. If det(**B**) can be ±1 the symbol is a pseudotensor.

It is convenient to relate det(**B**) to the metric tensor **g**. An element of **g**′ is given by (where parentheses indicate an inner product),

Take determinants,

Insert the positive value of det(**B**) into the transformation property of the Levi-Civita symbol,

then clearly the quantity η_{i1 i2...in} defined by

transforms as a covariant tensor. If det(**B**) is negative, η_{i1 i2...in} acquires an extra minus sign upon transformation, so that η_{i1 i2...in} is a pseudotensor.
For the record,

is a contravariant pseudotensor.

Let the inner product on *V* now be positive definite (and the space *V* be proper Euclidean) and consider only orthonormal bases. The matrix **B** transforming an orthonormal basis to another orthonormal basis, has the property **B**^{T}**B** = **I** (the identity matrix). Hence **B**^{T} = **B**^{−1}, i.e., **B** is an orthogonal matrix. From det(**B**^{T}) = det(**B**) = det(**B**^{−1}) = det(**B**)^{−1} follows that an orthogonal matrix has determinant ±1. Provided only orthogonal basis transformations are considered, the Levi-Civita symbol is either a tensor [if transformation are restricted to det(**B**)=1] or a pseudotensor [det(**B**)=−1 is also allowed]. The orthogonal transformations form a group, the orthogonal group in *n* dimensions, designated by O(n); its special [det(**B**)=1] subgroup is SO(n). The Levi-Civita symbol is an SO(n)-tensor (sometimes referred to as *Cartesian tensor*) and an O(n)-pseudotensor.