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ElMatrixThry - 4 Elementary Matrix Theory We will be using...

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Unformatted text preview: 4 Elementary Matrix Theory We will be using matrices: 1. For describing “change of basis” formulas. 2. For describing how to compute the result of any given linear operation acting on a “vector” (e.g., finding the components with respect to some basis of the force acting on an object having some velocity.) These are two completely different things. Do not confuse them even though the same computa- tional apparatus (i.e., matrices) is used for both. For example, if you confuse “rotating a vector” with “using a basis constructed by rotating the original basis”, you are likely to discover that your computations have everything spinning backwards. Throughout this set of notes, K , L , M and N are positive integers. 4.1 Basics Our Basic Notation A matrix A of size M × N is simply a rectangular array with M rows and N columns of “things”, A = a 11 a 12 a 13 ··· a 1 N a 21 a 22 a 23 ··· a 2 N a 31 a 32 a 33 ··· a 3 N . . . . . . . . . . . . . . . a M 1 a M 2 a M 3 ··· a MN . As indicated, I will try to use “bold face, upper case letters” to denote matrices. We will use two notations for the ( i , j ) th entry of A (i.e., the “thing” in the i th row and j th column): ( i , j ) th entry of A = a i j = [ A ] i j Until further notice, assume the “thing” in each entry is a scalar (i.e., a real or complex number). Later, we’ll use such “things” as functions, operators, and even other vectors and matrices. 9/11/2011 Elementary Matrix Theory Chapter & Page: 4–2 The matrix A is a row matrix if and only if it consists of just one row, and B is a column matrix if and only if it consists of just one column. In such cases we will normally simplify the indices in the obvious way, A = bracketleftbig a 1 a 2 a 3 ··· a N bracketrightbig and B = b 1 b 2 b 3 . . . b N . Basic Algebra Presumably, you are already acquainted with matrix equality, addition and multiplication. So all we’ll do here is express those concepts using our [·] i j notation: Matrix Equality Two matrices A and B are equal (and we write A = B ) if and only if both of the following hold: (a) A is the same size as B . (b) Letting M × N be the size of A and B , [ A ] jk = [ B ] jk for j = 1 . . . M and k = 1 . . . N . Matrix Addition and Scalar Multiplication Assuming A and B are both M × N matrices, and α and β are scalars, then α A + β B is the M × N matrix with entries [ α A + β B ] jk = α [ A ] jk + β [ B ] jk for j = 1 . . . M and k = 1 . . . N . Matrix Multiplication Assuming A is a L × M matrix and B is a M × N matrix, their product AB is the L × N matrix with entries [ AB ] jk = “ j th row of A times k th column of B ” = bracketleftbig a j 1 a j 2 a j 3 ··· a j M bracketrightbig b 1 k b 2 k b 3 k ....
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