S-Matrix Primer

Abstract

An m × n matrix is a rectangular array of numbers with m rows • and n columns. Example of a 2 × 3 matrix: � 3 1 4 � A =. 1 5 9 A ij denotes the number on row i and column j-for example, • A 13 = 4. The transpose of a matrix, denoted by a superscripted t, is a matrix • A t with the rows and columns interchanged, i.e., ij = A ji. For example, � 3 1 4 � t ⎛ 3 1 ⎞ 1 5 9 = ⎝ 4 1 9 5 ⎠ , Two matrices of identical shape can be added by adding their corre­ • sponding elements: If C = A+B, then C ij = A ij +B ij. Example: � 1 2 � � 10 20 � � 11 22 � + = 3 4 30 40 33 44 A matrix can be multiplied by a number by multiplying all of its • elements by that number: If B = aA, then B ij = aA ij. Example: � 1 2 � � 10 20 � 10 = × 3 4 30 40 The product C = AB of an l × m matrix A and an m × n matrix • B is defined as m C ij ≡ � A ik B kj. k=1