Handout 1 Elements of Matrix Algebra

Handout 1 Elements of Matrix Algebra - Handout

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August 23, 2010 Introduction to Matrix Algebra Definitions: Elementary Matrices One of the most important forms of algebra used in statistics is matrix algebra, or more generally, linear algebra. A matrix is simply an array of numbers. Here are three examples of matrices: A = 1 2 5 3 B = 4 3 1 5 2 5 C = 4 2 3 1 The numbers in the matrices are the elements of the matrix. The elements are identified by their rows first, then columns. Connecting the general forms below to the specific matrices above, we note that a 11 = 2, b 21 = 3, b 32 = 5 and c 31 = 2. A (2x2) = 21 11 a a 22 12 a a B (3x2) = 31 21 11 b b b 32 22 12 b b b C (4x1) = 41 31 21 11 c c c c The dimension of a matrix refers to the number of its rows and columns. Matrix A has dimension (2x2) and is designated A (2x2) whereas B has dimension (3x2) and is designated B (3x2) . In general if any matrix M has n – rows and k – columns, we denote it as M (nxk) . Matrices with one column (e.g., matrix C, above) are referred to as “vectors.” If you interchange the rows and columns of a matrix, you have its transpose . The transposes of the matrix B and the vector C, denoted as B T and C T , respectively, are: B T (2x3) = 5 2 5 4 3 1 C T = [ 1 3 2 4] A square matrix has the same number of columns as rows. Matrix A (above) is a square matrix. If, for each off- diagonal element of a square matrix, a ij = a ji , then the matrix is said to be a symmetric matrix . For example, the matrix S (3x3) = 6 5 3 5 4 2 3 2 1 is symmetric because s 21 = s 12 = 2, s 31 = s 13 = 3 and s 32 = s 23 = 5. Notice that symmetric matrix (S) is equal to its transpose (S T ). Two special symmetric matrices are the variance-covariance matrix (S) and the correlation matrix (R). In a square matrix such as S, the elements s 11 , s 22 and s 33 are the diagonal elements. A square matrix where all the off-diagonal elements are zero is a diagonal matrix . An example is the following diagonal matrix, D:
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D (3x3) = 6 0 0 0 4 0 0 0 1 A special diagonal matrix is the identity matrix , where all the diagonal elements are equal to 1. Identity matrices often function like “1’s” in scalar algebra (i.e., 1*4 = 1; 4/1 = 4; 4 * 1/4 = 1). The following is a (3x3) identity matrix: I (3x3) = 1 0 0 0 1 0 0 0 1 Elementary Matrix Operations: Addition and Multiplication Matrices can be added and subtracted, multiplied and divided, but only if certain conditions are met. To
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Handout 1 Elements of Matrix Algebra - Handout

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