7.2: Matrix and Index Notation

A vector can be described by listing its components along the \(xyz\) cartesian axes; for instance the displacement vector \(u\) can be denoted as \(u_x, u_y, u_z\), using letter subscripts to indicate the individual components. The subscripts can employ numerical indices as well, with 1, 2, and 3 indicating the \(x\), \(y\), and \(z\) directions; the displacement vector can therefore be written equivalently as \(u_1, u_2, u_3\).

A common and useful shorthand is simply to write the displacement vector as \(u_i\), where the \(i\) subscript is an index that is assumed to range over 1,2,3 ( or simply 1 and 2 if the problem is a two-dimensional one). This is called the range convention for index notation. Using the range convention, the vector equation \(u_i = a\) implies three separate scalar equations:

\[\begin{array} {c} {u_1 = a} \\ {u_2 = a} \\ {u_3 = a}\end{array} \nonumber\]

We will often find it convenient to denote a vector by listing its components in a vertical list enclosed in braces, and this form will help us keep track of matrix-vector multiplications a bit more easily. We therefore have the following equivalent forms of vector notation:

\[u = u_i = \left \{ \begin{matrix} u_1 \\ u_2 \\ u_3 \end{matrix} \right \} = \left \{ \begin{matrix} u_x \\ u_y \\ u_z \end{matrix} \right \} \nonumber\]

Second-rank quantities such as stress, strain, moment of inertia, and curvature can be denoted as \(3\times 3\) matrix arrays; for instance the stress can be written using numerical indices as

\[[\sigma] = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix}\nonumber\]

Here the first subscript index denotes the row and the second the column. The indices also have a physical meaning, for instance \(\sigma_{23}\) indicates the stress on the 2 face (the plane whose normal is in the 2, or \(y\), direction) and acting in the 3, or \(z\), direction. To help distinguish them, we'll use brackets for second-rank tensors and braces for vectors.

Using the range convention for index notation, the stress can also be written as \(\sigma_{ij}\), where both the \(i\) and the \(j\) range from 1 to 3; this gives the nine components listed explicitly above. (Since the stress matrix is symmetric, i.e. \(\sigma_{ij} = \sigma_{ji}\), only six of these nine components are independent.)

A subscript that is repeated in a given term is understood to imply summation over the range of the repeated subscript; this is the summation convention for index notation. For instance, to indicate the sum of the diagonal elements of the stress matrix we can write:

\[\sigma_{kk} = \sum_{k = 1}^{3} \sigma_{kk} = \sigma_{11} + \sigma_{22} + \sigma_{33}\nonumber\]

The multiplication rule for matrices can be stated formally by taking \(A =(a_{ij})\) to be an \((M \times N)\) matrix and \(B = (b_{ij})\) to be an \((R \times P)\) matrix. The matrix product \(AB\) is defined only when \(R = N\), and is the (\(M \times P\)) matrix \(C = (c_{ij})\) given by

\[c_{ij} = \sum_{k = 1}^{N} a_{ik} b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{iN} b_{Nk}\nonumber\]

Using the summation convention, this can be written simply

\[c_{ij} = a_{ik} b_{kj}\nonumber\]

where the summation is understood to be over the repeated index \(k\). In the case of a \(3 \times 3\) matrix multiplying a \(3 \times 1\) column vector we have

\[\begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix} \left \{ \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right \} = \left \{ \begin{matrix} a_{11}b_1 + a_{12}b_2 + a_{13} b_3 \\ a_{21}b_1 + a_{22}b_2 + a_{23} b_3 \\ a_{31}b_1 + a_{32}b_2 + a_{33} b_3 \end{matrix} \right \} = a_{ij} b_{j}\nonumber\]

The comma convention uses a subscript comma to imply differentiation with respect to the uses all of the variable following, so \(f_{,2} = \partial f/\partial y\) and \(u_{i,j} = \partial u_i/\partial x_j\). For instance, the expression \(\sigma_{ij, j} = 0\) three previously defined index conventions: range on \(i\), sum on \(j\), and differentiate:

\[\begin{array} {c} {\dfrac{\partial \sigma_{xx}}{\partial x} + \dfrac{\partial \sigma_{xy}}{\partial y} + \dfrac{\partial \sigma_{xz}}{\partial z} = 0} \\ {\dfrac{\partial \sigma_{yx}}{\partial x} + \dfrac{\partial \sigma_{yy}}{\partial y} + \dfrac{\partial \sigma_{yz}}{\partial z} = 0} \\ {\dfrac{\partial \sigma_{zx}}{\partial x} + \dfrac{\partial \sigma_{zy}}{\partial y} + \dfrac{\partial \sigma_{zz}}{\partial z} = 0} \end{array} \nonumber\]

The Kroenecker delta is a useful entity is defined as

\[\delta_{ij} = \begin{cases} 0, \ \ i \ne j \\ 1, \ \ i = j \end{cases} \nonumber\]

This is the index form of the unit matrix \(I\):

\[\delta_{ij} = I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\nonumber\]

So, for instance

\[\sigma_{kk}\delta_{ij} = \begin{bmatrix} \sigma_{kk} & 0 & 0 \\ 0 & \sigma_{kk} & 0 \\ 0 & 0 & \sigma_{kk} \end{bmatrix}\nonumber\]

where \(\sigma_{kk} = \sigma_{11} + \sigma_{22} + \sigma_{33}\).