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 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 \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. Vectors are "contravariant first-rank tensors". Contravariant tensors have their indices written as superscripts, while covariant tensors have their indices written as subscripts. So, the components of the above vector could be written ${\vec {\boldsymbol {x}}}=(x^{1},x^{2},x^{3})$. But doesn't writing the vector components as superscripts lead to ambiguity with raising something to a power? Not really. Proper tensor/vector equations almost never raise a vector component to a power. When one must do so, the convention is to put the power in parentheses. So $x^{2}$ is the second component of a vector, $x^{(2)}$ is the number x squared, and $x^{2^{(2)}}$ or $(x^{2})^{2}$ is the second component squared.

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:

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

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

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_{3j}x_j = b_3 }

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

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_{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 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 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.

REALLY CORRECT TENSOR INDEX NOTATION We can rewrite the preceding paragraphs in strict tensor form, by using the convention that if 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 \boldsymbol{A}} is a matrix, then 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 \boldsymbol{A}^i_j} is the element of that matrix in the ith row and jth column. So the equations, in terms of the correct vector components are: 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^1_1 x^1+a^1_2 x^2+a^1_3 x^3 = b^1 } 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^2_1 x^1+a^2_2 x^2+a^2_3 x^3 = b^2 } 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^3_1 x^1+a^3_2 x^2+a^3_3 x^3 = b^3 } Or, more succinctly: 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^1_j x^j = b^1 } 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^2_j x^j = b^2 } 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^3_j x^j = b^3 } Or just 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^i_j x^j = b^i } for all 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 i} from 1 to 3.

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:

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_{ij}x_j=b_i }

can be written simply 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}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 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}=b_{ik}c_{kl}d_{lj}} is valid but 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}=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 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}x_j + b_{ij}x_j} is valid but 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}x_j + b_{lj}x_j} is not as the first term does not have the index l.

REALLY CORRECT TENSOR INDEX NOTATION The rules for really correct tensor notation, including Einstein Summation, are as follows: Einstein summation must be over one subscript and one superscript. Indices that appear in terms to be added or subtracted, or on opposite sides of and equals sign, must match: subscript for subscript, and superscript for superscript. For any index left over after summation, there is an implicit "for all" in the equation. So the above equation becomes just 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 b^i = a^i_j x^j } That is the really proper way to express that vector 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 \vec{\boldsymbol{b}}} is the result of applying the linear transformation 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 \boldsymbol{A}} to the vector 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 \vec{\boldsymbol{x}}}

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

See also

• Tensor