- Einstein notation
In

mathematics , especially in applications oflinear algebra tophysics , the**Einstein notation**or**Einstein summation convention**is a notational convention useful when dealing with coordinate formulas. It was introduced byAlbert Einstein in 1916. cite journal| last = Einstein| first = Albert| authorlink = Albert Einstein| coauthors = | title = The Foundation of the General Theory of Relativity| journal = Annalen der Physik| volume = | issue = | pages = | date = 1916| publisher = | url = http://www.alberteinstein.info/gallery/gtext3.html| format =PDF | id = | accessdate = 2006-09-03 ]According to this convention, when an index variable appears twice in a single term, once in an upper (superscript) and once in a lower (subscript) position, it implies that we are summing over all of its possible values. In typical applications, the indices are 1,2,3 (representing the three dimensions of physical

Euclidean space ), or 0,1,2,3 or 1,2,3,4 (representing the four dimensions of space-time, orMinkowski space ), but they can have any range, even (in some applications) aninfinite set . Thus in three dimensions:$y\; =\; c\_i\; x^i\; ,$

actually means

:$y\; =\; sum\_\{i=1\}^3\; c\_i\; x\_i$.

Abstract index notation is an improvement of Einstein notation.In

general relativity , theGreek alphabet and theRoman alphabet are used to distinguish whether summing over 1,2,3 or 0,1,2,3 (usually Roman, "i", "j", ... for 1,2,3 and Greek, "$mu,$", $u,$, ... for 0,1,2,3). As in sign conventions, the convention used in practice varies: Roman and Greek may be reversed.When there is a fixed basis, one can work with only subscripts,but in general one must distinguish between superscripts and subscripts;

see below.It is important to keep in mind that no new physical laws or ideas result from using Einstein notation; rather, it merely helps in identifying relationships and symmetries often 'hidden' by more conventional notation.

In some fields, Einstein notation is referred to simply as index notation, or indicial notation.The use of the implied summation of repeated indices is also referred to as the "Einstein Sum Convention."

**Introduction**The basic idea of Einstein notation is very simple. It allows one to replace something bulky, such as:

:$y\; =\; c\_1x\_1+c\_2x\_2+c\_3x\_3+\; ...\; +\; c\_nx\_n\; ,$

typically written as:

:$y\; =\; sum\_\{i=1\}^n\; c\_ix\_i$

with something even simpler, in "Einstein notation":

:$y\; =\; c\_i\; x^i\; ,$

In Einstein notation, indices such as "i" in the equation above can appear as either subscripts or superscripts. The position of the index has a specific meaning. It is important, of course, not to interpret an index appearing in the superscript position as if it were an exponent, which is the convention in standard algebra. Here, the superscripted "i" above the symbol "x" represents an integer-valued index running from 1 to "n".

The virtue of Einstein notation is that an index appearing two or more times in a single term implies summation across that index, so that the summation symbol is unnecessary. Since the summation in effect "eliminates" the index over which the sum is taken, the summation index does not appear on the opposite side of the equals sign.

**Vector representations**First, we can use Einstein notation in

linear algebra to distinguish easily between vectors and covectors: upper indices represent the "components" of vectors,while lower indices represent the "components" of covectors.However, vectors themselves (not their components) have lower indices,and covectors have upper indices.:

**"This point is frequently confused.**"Given a vector space "V" and its

dual space $V^*$,one represents vectors (elements of "V") with subscripts, as in $v\_i\; in\; V$, andcovector s with superscripts, as in $w^i\; in\; V^*$.However, the "components" of vectors and covectors follow the opposite convention:if $e\_i$ are a basis for "V" and $e^i$ are the dual basis for $V^*$, then vectors are represented as::$v\; =\; a^i\; e\_i\; =\; egin\{bmatrix\}a^1\backslash a^2\backslash vdots\backslash a^nend\{bmatrix\}$and covectors are represented as:$w\; =\; a\_i\; e^i\; =\; egin\{bmatrix\}a\_1\; a\_2\; cdots\; a\_nend\{bmatrix\}$This is because a component of a vector (a coefficient in some basis) is the "value" of a "covector": the coefficient of $e\_i$ is the value of the corresponding covector in the

dual basis : $a^i\; =\; e^i(v)$.Note that $e^i$ is a covector, but $a^i$ is a scalar.More prosaically, you pair components with vectors; since vectors have lower indices,components have upper indices.In terms of

covariance and contravariance of vectors , lower indices represent (components of!) covariant vectors (covector s), while upper indices represent (components of!)contravariant vectors (vectors): they transform covariantly (resp., contravariantly) with respect to change of coordinates.A particularly confusing notation is to use the same letter both for a (co)vector and its components, as in::$v\; =\; v^i\; e\_i\; =\; egin\{bmatrix\}v^1\backslash v^2\backslash vdots\backslash v^nend\{bmatrix\}$:$w\; =\; w\_i\; e^i\; =\; egin\{bmatrix\}w\_1\; w\_2\; cdots\; w\_nend\{bmatrix\}$Here $v^i$ does not mean "the covector "v", but rather,"the components of the vector "v".

**Mnemonics*** "

**Up**per indices go**up**to down;**l**ower indices go**l**eft to right"

* You can stack vectors (column matrices) side-by-side::$egin\{bmatrix\}v\_1\; cdots\; v\_kend\{bmatrix\}.$Hence the lower index indicates which "column" you are in.

* You can stack covectors (row matrices) top-to-bottom::$egin\{bmatrix\}w^1\; \backslash \; vdots\; \backslash \; w^kend\{bmatrix\}$Hence the upper index indicates which "row" you are in.**Superscripts and subscripts vs. only subscripts**In the presence of a non-degenerate form (an isomorphism $V\; o\; V^*$),(for instance a

Riemannian metric orMinkowski metric ),one can raise and lower indices.A basis gives such a form (via the

dual basis ), hence when working on$mathbf\{R\}^n$ with a fixed basis, one can work with just subscripts.However, if one changes coordinates, the way that coefficients change depends on the variance of the object, and one cannot ignore the distinction;see

covariance and contravariance of vectors .**Common operations in this notation**In Einstein notation, the usual element reference $mathbf\{A\}\_\{mn\}$ for the $m$th row and $n$th column of matrix $mathbf\{A\}$ becomes $mathbf\{A\}\_n^m$. We can then write the following operations in Einstein notation as follows.

**Inner product**Given a row vector $v\_i$ and a column vector $u^i$ of the same size, we can take the

inner product $v\_i\; u^i$, which is a scalar: it's evaluating the covector on the vector.**Multiplication of a vector by a matrix**Given a matrix $A^i\_j$ and a (column) vector $v^j$, the coefficients of the product $mathbf\{A\}v$ are given by $A^i\_j\; v^j$.

**Matrix multiplication**We can represent

matrix multiplication as::$C^i\_k\; =\; A^i\_j\; cdot\; B^j\_k$

This expression is equivalent to the more conventional (and less compact) notation:

:$mathbf\{C\}\_\{ik\}\; =\; (mathbf\{A\}\; cdot\; mathbf\{B\})\_\{ik\}\; =sum\_\{j=1\}^N\; A\_\{ij\}\; B\_\{jk\}$

**Trace**Given a matrix $A^i\_j$, summing over a common index $A^i\_i$ yields the trace.

**Outer product**The

outer product of the column vector**u**by the row vector**v**yields an "M" × "N" matrix**A**::$mathbf\{A\}\; =\; mathbf\{u\}\; cdot\; mathbf\{v\}$

In Einstein notation, we have:

:$A^i\_j\; =\; u^i\; cdot\; v\_j\; =\; (uv)^i\_j$

Since "i" and "j" represent two "different" indices, and in this case over two different ranges "M" and "N" respectively, the indices are not eliminated by the multiplication. Both indices survive the multiplication to become the two indices of the newly-created matrix "A".

**Coefficients on tensors and related**Given a

tensor field and a basis (of linearly independent vector fields),the coefficients of the tensor field in a basis can be computed by evaluating on a suitable combination of the basis and dual basis, and inherits the correct indexing.We list notable examples.Throughout, let $e\_i$ be a basis of vector fields (a

moving frame ).* (covariant)

metric tensor :$g\_\{ij\}\; =\; g(e\_i,e\_j)$

* (contravariant)metric tensor :$g^\{ij\}\; =\; g(e^i,e^j)$

*Torsion tensor (using the below):$T^c\_\{ab\}\; =\; Gamma^c\_\{ab\}\; -\; Gamma^c\_\{ba\}-gamma^c\_\{ab\},$which follows from the formula:$T\; =\; abla\_X\; Y\; -\; abla\_Y\; X\; -\; [X,Y]\; .$

*Riemann curvature tensor :$\{R^\; ho\}\_\{sigmamu\; u\}\; =\; dx^\; ho(R(partial\_\{mu\},partial\_\{\; u\})partial\_\{sigma\})$This also applies for some operations that are not

tensorial , for instance:

*Christoffel symbol s:$abla\_ie\_j=Gamma\_\{ij\}^ke\_k$where $abla\_i\; e\_j$ is thecovariant derivative .Equivalently,:$Gamma\_\{ij\}^k\; =\; e^k\; abla\_ie\_j$

* commutator coefficients:$[e\_i,e\_j]\; =\; gamma\_\{ij\}^k\; e\_k$where $[e\_i,e\_j]$ is theLie bracket .Equivalently,:$gamma\_\{ij\}^k\; =\; e^k\; [e\_i,e\_j]\; .$**Vector dot product**In mechanics and engineering, vectors in 3D space are often described in relation to orthogonal

unit vector s**i**,**j**and**k**.:$mathbf\{u\}\; =\; u\_x\; mathbf\{i\}\; +\; u\_y\; mathbf\{j\}\; +\; u\_z\; mathbf\{k\}$

If the basis vectors

**i**,**j**, and**k**are instead expressed as**e**_{1},**e**_{2}, and**e**_{3}, a vector can be expressed in terms of a summation::$mathbf\{u\}\; =\; u^1\; mathbf\{e\}\_1\; +\; u^2\; mathbf\{e\}\_2\; +\; u^3\; mathbf\{e\}\_3\; =\; sum\_\{i\; =\; 1\}^3\; u^i\; mathbf\{e\}\_i$

In Einstein notation, the summation symbol is omitted since the index " i " is repeated once as an upper index and once as a lower index, and we simply write

:$mathbf\{u\}\; =\; u^i\; mathbf\{e\}\_i$

Using

**e**_{1},**e**_{2}, and**e**_{3}instead of**i**,**j**, and**k**, together with Einstein notation, we obtain a concise algebraic presentation of vector andtensor equations. For example,:$mathbf\{u\}\; cdot\; mathbf\{v\}\; =\; left(\; sum\_\{i\; =\; 1\}^3\; u^i\; mathbf\{e\}\_i\; ight)\; cdot\; left(\; sum\_\{j\; =\; 1\}^3\; v^j\; mathbf\{e\}\_j\; ight)\; =\; (u^i\; mathbf\{e\}\_i)\; cdot\; (v^j\; mathbf\{e\}\_j)=\; u^i\; v^j\; (\; mathbf\{e\}\_i\; cdot\; mathbf\{e\}\_j\; ).$

Since:$mathbf\{e\}\_i\; cdot\; mathbf\{e\}\_j\; =\; delta\_\{ij\}$

where $delta\_\{ij\}$ is the

Kronecker delta , which is equal to 1 when "i" = "j", and 0 otherwise, we find:$mathbf\{u\}\; cdot\; mathbf\{v\}\; =\; u^i\; v^jdelta\_\{ij\}.$One can use $delta\_\{ij\}$ to lower indices of the vectors; namely, $u\_i=delta\_\{ij\}u^j$ and $v\_i=delta\_\{ij\}v^j$. Then:$mathbf\{u\}\; cdot\; mathbf\{v\}\; =\; u^i\; v^jdelta\_\{ij\}=\; u^i\; v\_i\; =\; u\_j\; v^j$Note that, despite $u^i=u\_i$ for any fixed $i$, it is incorrect to write:$mathbf\{u\}\; cdot\; mathbf\{v\}\; =\; u^iv^i,$since on the right hand side the index $i$ is repeated both times as an upper index and so there is no summation over $i$ according to the Einstein convention. Rather, one should explicitly write the summation::$mathbf\{u\}\; cdot\; mathbf\{v\}\; =\; sum\_\{i=1\}^3u^iv^i.$**Vector cross product**For the

cross product ,:$mathbf\{u\}\; imes\; mathbf\{v\}=\; left(\; sum\_\{j\; =\; 1\}^3\; u^j\; mathbf\{e\}\_j\; ight)\; imes\; left(\; sum\_\{k\; =\; 1\}^3\; v^k\; mathbf\{e\}\_k\; ight)\; =\; (u^j\; mathbf\{e\}\_j\; )\; imes\; (v^k\; mathbf\{e\}\_k)$:$=\; u^j\; v^k\; (mathbf\{e\}\_j\; imes\; mathbf\{e\}\_k\; )\; =\; u^j\; v^kepsilon^i\_\{jk\}\; mathbf\{e\}\_i$

where $mathbf\{e\}\_j\; imes\; mathbf\{e\}\_k\; =\; epsilon^i\_\{jk\}\; mathbf\{e\}\_i$ and $epsilon^i\_\{jk\}=delta^\{il\}epsilon\_\{ljk\}$, with $epsilon\_\{ijk\}$ the

Levi-Civita symbol defined by::$epsilon\_\{ijk\}\; =left\{egin\{matrix\}0\; mbox\{unless\; \}\; i,j,k\; mbox\{\; are\; distinct\}\backslash +1\; mbox\{if\; \}\; (i,j,k)\; mbox\{\; is\; an\; even\; permutation\; of\; \}\; (1,2,3)\backslash -1\; mbox\{if\; \}\; (i,j,k)\; mbox\{\; is\; an\; odd\; permutation\; of\; \}\; (1,2,3)end\{matrix\}\; ight.$

One then recovers

:$mathbf\{u\}\; imes\; mathbf\{v\}\; =\; (u^2\; v^3\; -\; u^3\; v^2)\; mathbf\{e\}\_1\; +\; (u^3\; v^1\; -\; u^1\; v^3)\; mathbf\{e\}\_2\; +\; (u^1\; v^2\; -\; u^2\; v^1)\; mathbf\{e\}\_3$

from

:$mathbf\{u\}\; imes\; mathbf\{v\}=\; epsilon^i\_\{jk\}\; u^j\; v^kmathbf\{e\}\_i\; =\; sum\_\{i\; =\; 1\}^3\; sum\_\{j\; =\; 1\}^3\; sum\_\{k\; =\; 1\}^3\; epsilon^i\_\{jk\}\; u^j\; v^kmathbf\{e\}\_i$.In other words, if $mathbf\{w\}\; =\; mathbf\{u\}\; imes\; mathbf\{v\}$, then $w^i\; mathbf\{e\}\_i=\; epsilon^i\_\{jk\}\; u^j\; v^kmathbf\{e\}\_i$, so that $w^i\; =\; epsilon^i\_\{jk\}\; u^j\; v^k$.

**Abstract definitions**In the traditional usage, one has in mind a

vector space "V" with finite dimension "n", and a specific basis of "V". We can write the basis vectors as**e**_{1},**e**_{2}, ...,**e**_{"n"}. Then if**v**is a vector in "V", it has coordinates $v^1,dots,v^n$ relative to this basis.The basic rule is:: $mathbf\{v\}\; =\; v^imathbf\{e\}\_i.$In this expression, it was assumed that the term on the right side was to be summed as "i" goes from 1 to "n", because the index "i" does not appear on both sides of the expression. (Or, using Einstein's convention, because the index "i" appeared twice.)

The "i" is known as a "dummy index" since the result is not dependent on it; thus we could also write, for example:: $mathbf\{v\}\; =\; v^jmathbf\{e\}\_j.$An index that is not summed over is a "free index" and should be found in each term of the equation or formula. Compare dummy indices and free indices with

free variables and bound variables .The value of the Einstein convention is that it applies to other vector spaces built from "V" using the

tensor product and duality. For example, $Votimes\; V$, the tensor product of "V" with itself, has a basis consisting of tensors of the form $mathbf\{e\}\_\{ij\}\; =\; mathbf\{e\}\_i\; otimes\; mathbf\{e\}\_j$. Any tensor**T**in $Votimes\; V$ can be written as::$mathbf\{T\}\; =\; T^\{ij\}mathbf\{e\}\_\{ij\}$."V*", the dual of "V", has a basis

**e**^{1},**e**^{2}, ...,**e**^{"n"}which obeys the rule:$mathbf\{e\}^i\; (mathbf\{e\}\_j)\; =\; delta^i\_j$.Here δ is theKronecker delta , so $delta^i\_j$ is 1 if "i" ="j" and 0 otherwise.As:$mathrm\{Hom\}(V,W)\; =\; V^*\; otimes\; W$the row-column coordinates on a matrix correspond to the upper-lower indices on the tensor product.

**Examples**Einstein summation is clarified with the help of a few simple examples. Consider four-dimensional spacetime, where indices run from 0 to 3:

:$mathbf\{\}\; a^mu\; b\_mu\; =\; a^0\; b\_0\; +\; a^1\; b\_1\; +\; a^2\; b\_2\; +\; a^3\; b\_3$

:$mathbf\{\}\; a^\{mu\; u\}\; b\_mu\; =\; a^\{0\; u\}\; b\_0\; +\; a^\{1\; u\}\; b\_1\; +\; a^\{2\; u\}\; b\_2\; +\; a^\{3\; u\}\; b\_3.$

The above example is one of contraction, a common tensor operation. The tensor $mathbf\{\}\; a^\{mu\; u\}b\_\{mu\}$ becomes a new tensor by summing over the first upper index and the lower index. Typically the resulting tensor is renamed with the contracted indices removed:

:$mathbf\{\}\; \{s\}^\{\; u\}\; =\; a^\{mu\; u\}b\_\{mu\}.$

For a familiar example, consider the dot product of two vectors

**a**and**b**. The dot product is defined simply as summation over the indices of**a**and**b**::$mathbf\{a\}cdotmathbf\{b\}\; =\; a^\{alpha\}b\_\{alpha\}\; =\; a^0\; b\_0\; +\; a^1\; b\_1\; +\; a^2\; b\_2\; +\; a^3\; b\_3,$

which is our familiar formula for the vector dot product. Remember it is sometimes necessary to change the components of

**a**in order to lower its index; however, this is not necessary in Euclidean space, or any space with a metric equal to its inverse metric (e.g., flat spacetime).**See also***

Abstract index notation

*Bra-ket notation

*Penrose graphical notation **References*** cite news

last=Rawlings

first=Steve

url=http://www-astro.physics.ox.ac.uk/~sr/lectures/vectors/lecture10final.pdfc

title=Lecture 10 - Einstein Summation Convention and Vector Identities

publisher=Oxford University

date=2007-02-01

*Wikimedia Foundation.
2010.*

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