Tensor index notation

Tensor index notation is a method of notation which is of use when dealing with tensors. Particular examples of tensors include vectors and matrices, and index notation can greatly simplify algebraic manipulations involving these types of mathematical object.

The components of a vector (with respect to some co-ordinate system) might be written ${\vec {\boldsymbol {x}}}=(x_{1},x_{2},x_{3})$.

REALLY CORRECT TENSOR INDEX NOTATION While elementary treatments of linear algebra and matrix algebra are normally done with subscripts everywhere, proper tensor index notation involves subscripts and superscripts placed very carefully. While the exact reasons for this are beyond the scope of these articles, the articles will show what correct placement looks like. Proper index notation is actually simpler once one gets used to it and understands it. For now, the fundamental principle, is that: Indices of vectors are written as superscripts. Get used to seeing things written that way. So, the components of the above vector could be written ${\vec {\boldsymbol {x}}}=(x^{1},x^{2},x^{3})$.

More concisely, we could write $x_{i}$ for the components of the vector, where $i=1,2,3$. To motivate this notation, we will consider the equation ${\boldsymbol {A}}\,{\vec {\boldsymbol {x}}}={\vec {\boldsymbol {b}}}$ for some matrix ${\boldsymbol {A}}$ and vectors ${\vec {\boldsymbol {x}}}$ and ${\vec {\boldsymbol {b}}}$. We will use the convention that if ${\boldsymbol {A}}$ is a matrix, then $({\boldsymbol {A}})_{ij}=a_{ij}$ is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns $x_{1},x_{2},x_{3}$:

$a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}=b_{1}$

$a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}=b_{2}$

$a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}=b_{3}$

This can be rewritten in a matrix/vector form as equation ${\boldsymbol {A}}\,{\vec {\boldsymbol {x}}}={\vec {\boldsymbol {b}}}$:

${\begin{pmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\end{pmatrix}}$

Comparison of these two forms should convince you that the "go along the column and down the rows" rule for multiplying a matrix and a vector is sensible. We can also write the above system of equations more succinctly in index notation. We notice that in any of the three equations, the first index on the $a_{ij}$ elements is fixed whilst the second varies from 1 to 3. Thus:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{j=1}^3 a_{1j}x_j = b_1 }

$\sum _{j=1}^{3}a_{2j}x_{j}=b_{2}$

$\sum _{j=1}^{3}a_{3j}x_{j}=b_{3}$

Even more succinctly, we can write this as the single expression

$\sum _{j=1}^{3}a_{ij}x_{j}=b_{i}$

When you see such an equation, remember that it is a shorthand notation for writing three equations at once, for $i=1,2,3$ (in 3D). The major advantage with index notation is that objects such as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{ij}} as scalars and therefore commute (ab=ba). Tensors themselves however can represent objects such as matrices and do not in general commute.

Einstein Summation Convention

Index notation greatly simplifies the manipulation of tensors. A commonly used convention when dealing with tensors is the so called "Einstein summation convention." Any index that is repeated exactly twice is assumed to be summed over and the sum symbols are not written. In the example above, the sum:

$\sum _{j=1}^{3}a_{ij}x_{j}=b_{i}$

can be written simply as $a_{ij}x_{j}=b_{i}$. There are three rules when using this convention:

Any repeated index is implicitly summed over, as above.

Each index cannot appear more than twice in any term, so $a_{ij}=b_{ik}c_{kl}d_{lj}$ is valid but $a_{ij}=b_{ik}c_{kk}d_{kj}$ is not as it has more than two k's.

Each term must contain the same non-repeated indices, so $a_{ij}x_{j}+b_{ij}x_{j}$ is valid but $a_{ij}x_{j}+b_{lj}x_{j}$ is not as the first term does not have the index l.

In general, the indices may subscript or superscript depending on whether the tensors are covariant or contravariant, respectively.

See also

• Tensor