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AppMatrix - Introduction to Financial Econometrics Appendix...

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IntroductiontoFinancialEconometrics Appendix MatrixAlgebraReview EricZivot DepartmentofEconomics UniversityofWashington January3,2000 Thisversion:February6,2001 1 Matrix Algebra Review A matrix isjustanarrayofnumbers. The dimension ofamatrixisdeterminedby thenumberofitsrowsandcolumns. Forexample,amatrix A with n rowsand m columnsisillustratedbelow A ( n × m ) = a 11 a 12 . . . a 1 m a 21 a 22 . . . a 2 m . . . . . . . . . . . . a n 1 a n 2 . . . a nm where a ij denotesthe i th rowand j th columnelementof A . A vector issimplyamatrixwith 1 column.Forexample, x ( n × 1) = x 1 x 2 . . . x n isan n × 1 vectorwithelements x 1 , x 2 , . . . , x n . Vectorsandmatricesareoftenwritten inbold type (or underlined) to distinguish them from scalars (single elements of vectorsormatrices). The transpose ofan n × m matrix A isanewmatrixwiththerowsandcolumns of A interchangedandisdenoted A 0 or A | . Forexample, A (2 × 3) = · 1 2 3 4 5 6 ¸ , A 0 (3 × 2) = 1 4 2 5 3 6 1
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x (3 × 1) = 1 2 3 , x 0 (1 × 3) = £ 1 2 3 ¤ . A symmetric matrix A issuchthat A = A 0 . Obviouslythiscanonlyoccurif A isa square matrix;i.e.,thenumberofrowsof A isequaltothenumberofcolumns. Forexample,considerthe 2 × 2 matrix A = · 1 2 2 1 ¸ . Clearly, A 0 = A = · 1 2 2 1 ¸ . 1.1 Basic Matrix Operations 1.1.1 Addition and subtraction Matrixadditionandsubtractionareelementbyelementoperationsandonlyapply tomatricesofthesamedimension.Forexample,let A = · 4 9 2 1 ¸ , B = · 2 0 0 7 ¸ . Then A + B = · 4 9 2 1 ¸ + · 2 0 0 7 ¸ = · 4 + 2 9 + 0 2 + 0 1 + 7 ¸ = · 6 9 2 8 ¸ , A B = · 4 9 2 1 ¸ · 2 0 0 7 ¸ = · 4 2 9 0 2 0 1 7 ¸ = · 2 9 2 6 ¸ . 1.1.2 Scalar Multiplication Herewerefertothemultiplicationofamatrixbyascalarnumber. Thisisalsoan element-by-elementoperation.Forexample,let c = 2 and A = · 3 1 0 5 ¸ . Then c · A = · 2 · 3 2 · ( 1) 2 · (0) 2 · 5 ¸ = · 6 2 0 10 ¸ . 2
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1.1.3 Matrix Multiplication Matrixmultiplicationonlyappliesto conformable matrices. A and B areconformable matricesofthenumberofcolumnsin A isequaltothenumberofrowsin B . For example,if A is m × n and B is m × p
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