Matrix and Index Notation David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 September 18 2000 A vector can be described by listing its components along the xyz cartesian axes for instance the displacement vector u can be denoted as ux uy uz 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 u1 u2 u3 A common and useful shorthand is simply to write the displacement vector as ui 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 ui a implies three separate scalar equations u1 a u2 a u3 a 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 ui u1 u 2 u 3 ux u y u z Second rank quantities such as stress strain moment of inertia and curvature can be denoted as 3 3 matrix arrays for instance the stress can be written using numerical indices as 11 12 13 21 22 23 31 32 33 Here the first subscript index denotes the row and the second the column The indices also have a physical meaning for instance 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 ij where both the i and the j range from 1 to 3 this gives the nine components listed explicitly above 1 Since the stress matrix is symmetric i e ij 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 kk 3 X kk 11 22 33 k 1 The multiplication rule for matrices can be stated formally by taking A aij to be an M N matrix and B bij to be an R P matrix The matrix product AB is defined only when R N and is the M P matrix C cij given by cij N X aik bkj ai1 b1j ai2 b2j aiN bN k k 1 Using the summation convention this can be written simply cij aik bkj where the summation is understood to be over the repeated index k In the case of a 3 3 matrix multiplying a 3 1 column vector we have a11 a12 a13 b1 b a21 a22 a23 2 a31 a32 a33 b3 a11 b1 a12 b2 a13 b3 a b a b a b 21 1 22 2 23 3 a b a b a b 31 1 32 2 33 3 aij bj The comma convention uses a subscript comma to imply differentiation with respect to the variable following so f 2 f y and ui j ui xj For instance the expression ij j 0 uses all of the three previously defined index conventions range on i sum on j and differentiate xx xy xz 0 x y z yx yy yz 0 x y z zx zy zz 0 x y z The Kroenecker delta is a useful entity is defined as ij 0 1 i 6 j i j This is the index form of the unit matrix I 1 0 0 ij I 0 1 0 0 0 1 So for instance 2 kk 0 0 kk ij 0 kk 0 0 0 kk where kk 11 22 33 3
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