Definition:Matrix/Summation Convention

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Definition

The summation convention is a notational device used in the manipulation of matrices, in particular square matrices in the context of physics and applied mathematics.

If the same index occurs twice in a given expression involving matrices, then summation over that index is automatically assumed.

Thus the summation sign can be omitted, and expressions can be written more compactly.


Examples

Trace of Matrix

The trace of $A$, using the summation convention, is:

$\tr \paren A = a_{ii}$


Substitution Rule

The Substitution Rule for Matrices can be expressed using the summation convention as:

$(1): \quad \delta_{i j} a_{j k} = a_{i k}$
$(2): \quad \delta_{i j} a_{k j} = a_{k i}$

where:

$\delta_{i j}$ is the Kronecker delta
$a_{j k}$ is element $\left({j, k}\right)$ of $\mathbf A$.


The index which appears twice in these expressions is the element $j$, which is the one summated over.


Determinant of Order 3

The determinant of a square matrix of order $3$ $\mathbf A$ can be expressed using the summation convention as:

$\det \left({\mathbf A}\right) = \dfrac 1 6 \operatorname{sgn} \left({i, j, k}\right) \operatorname{sgn} \left({r, s, t}\right) a_{i r} a_{j s} a_{k t}$


Note that there are six indices which appear twice, and so six summations are assumed.


Matrix Product

The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the summation convention as:

Then:

$c_{i j} := a_{i k} \circ b_{k j}$


The index which appears twice in the expressions on the right hand side is the entry $k$, which is the one summated over.


Sources