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Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions : A matrix is a collection of numbers ordered by rows and columns. It is customary to enclose the elements of a matrix in parentheses, brackets, or braces. For example, the following is a matrix: X = 5 8 2 - 1 0 7 . This matrix has two rows and three columns, so it is referred to as a “2 by 3” matrix. The elements of a matrix are numbered in the following way: X = x 11 x 12 x 13 x 21 x 22 x 23 That is, the first subscript in a matrix refers to the row and the second subscript refers to the column. It is important to remember this convention when matrix algebra is performed. A vector is a special type of matrix that has only one row (called a row vector ) or one column (called a column vector ). Below, a is a column vector while b is a row vector. a = 7 2 3 , b = - 2 7 4 ( 29 A scalar is a matrix with only one row and one column. It is customary to denote scalars by italicized, lower case letters (e.g., x ), to denote vectors by bold, lower case letters (e.g., x ), and to denote matrices with more than one row and one column by bold, upper case letters (e.g., X ). A square matrix has as many rows as it has columns. Matrix A is square but matrix B is not square: A = 1 6 3 2 , B = 1 9 0 3 7 - 2 A symmetric matrix is a square matrix in which x ij = x ji for all i and j . Matrix A is symmetric; matrix B is not symmetric. A = 9 1 5 1 6 2 5 2 7 , B = 9 1 5 2 6 2 5 1 7 A diagonal matrix is a symmetric matrix where all the off diagonal elements are 0. Matrix A is diagonal.

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Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 2 A = 9 0 0 0 6 0 0 0 7 An identity matrix is a diagonal matrix with 1s and only 1s on the diagonal. The identity matrix is almost always denoted as I . I = 1 0 0 0 1 0 0 0 1 Matrix Addition and Subtraction : To add two matrices, they both must have the same number of rows and they both must have the same number of columns. The elements of the two matrices are simply added together, element by element, to produce the results. That is, for R = A + B, then r ij = a ij + b ij for all i and j . Thus, 9 5 1 - 4 7 6 = 1 9 - 2 3 6 0 + 8 - 4 3 - 7 1 6 Matrix subtraction works in the same way, except that elements are subtracted instead of added. Matrix Multiplication : There are several rules for matrix multiplication. The first concerns the multiplication between a matrix and a scalar. Here, each element in the product matrix is simply the scalar multiplied by the element in the matrix. That is, for R = a B, then r ij = ab ij for all i and j . Thus, 8 2 6 3 7 = 16 48 24 56 Matrix multiplication involving a scalar is commutative. That is, a B = B a .
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