AppMatrix - Introduction to Financial Econometrics Appendix Matrix Algebra Review Eric Zivot Department of Economics University of Washington

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Introduction to Financial Econometrics Appendix Matrix Algebra Review Eric Zivot Department of Economics University of Washington January 3, 2000 This version: February 6, 2001 1 Matrix Algebra Review A matrix is just an array of numbers. The dimension of a matrix is determined by the number of its rows and columns. For example, a matrix A with n rows and m co lumnsisi l lustratedbe low A ( n × m ) = a 11 a 12 ... a 1 m a 21 a 22 2 m . . . . . . ... . . . a n 1 a n 2 nm where a ij denotes the i th row and j th column element of A . A vector is simply a matrix with 1 column. For example, x ( n × 1) = x 1 x 2 . . . x n is an n × 1 vector with elements x 1 ,x 2 ,...,x n . Vectors and matrices are often written in bold type (or underlined) to distinguish them from scalars (single elements of vectors or matrices). The transpose of an n × m matrix A is a new matrix with the rows and columns of A interchanged and is denoted A 0 or A | . For example, A (2 × 3) = · 123 456 ¸ , A 0 (3 × 2) = 14 25 36 1
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x (3 × 1) = 1 2 3 , x 0 (1 × 3) = £ 123 ¤ . A symmetric matrix A is such that A = A 0 . Obviously this can only occur if A is a square matrix; i.e., the number of rows of A is equal to the number of columns. Forexamp le ,cons iderthe 2 × 2 matrix A = · 12 21 ¸ . Clearly, A 0 = A = · ¸ . 1.1 Basic Matrix Operations 1.1.1 Addition and subtraction Matrix addition and subtraction are element by element operations and only apply to matrices of the same dimension. For example, let A = · 49 ¸ , B = · 20 07 ¸ . Then A + B = · ¸ + · ¸ = · 4+2 9+0 2+0 1+7 ¸ = · 69 28 ¸ , A B = · ¸ · ¸ = · 4 29 0 2 01 7 ¸ = · 2 6 ¸ . 1.1.2 Scalar Multiplication Here we refer to the multiplication of a matrix by a scalar number. This is also an element-by-element operation. For example, let c =2 and A = · 3 1 05 ¸ . Then c · A = · 2 · 32 · ( 1) 2 · (0) 2 · 5 ¸ = · 6 2 0 ¸ . 2
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1.1.3 Matrix Multiplication Matrix multiplication only applies to conformable matrices. A and B are conformable matrices of the number of columns in A is equal to the number of rows in B .
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This note was uploaded on 09/13/2011 for the course ECON 503 taught by Professor Pujara during the Spring '11 term at Punjab Engineering College.

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AppMatrix - Introduction to Financial Econometrics Appendix Matrix Algebra Review Eric Zivot Department of Economics University of Washington

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