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Lecture26-2010 - Review of Matrix Algebra and Vector Spaces...

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Review of Matrix Algebra and Vector Spaces Charles B. Moss August 3, 2010 I. Review of Elementary Matrix Algebra A. Basic Definitions 1. A matrix A of size m × n is an m × n rectangular array of scalars A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 · · · a mn (1) It is sometimes useful to partition matrices into vectors. A = a · 1 a · 2 · · · a · n a · 1 = a 11 a 21 . . . a m 1 · · · a · n = a 1 n a 2 n . . . a mn (2) A = a 1 · a 2 · . . . a m · a 1 · = a 11 a 12 · · · a 1 n a 2 · = a 21 a 22 · · · a 2 n . . . a m · = a m 1 a m 2 · · · a mn (3) 2. The sum of two identically dimensioned matrices can be ex- pressed as 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVI Fall 2010 A + B = [ a ij + b ij ] (4) 3. In order to multiply a matrix by a scalar, multiply each ele- ment of the matrix by the scalar. 4. In order to discuss matrix multiplication, we first discuss vec- tor multiplication. Two vectors x and y can be multiplied together to form z ( z = x · y ) only if they are conformable. If x is of order 1 × n and y is of order n × 1, then the vectors are conformable and the multiplication becomes z = xy = n i =1 x i y i (5) Extending this discussion to matrices, two matrices A and B can be multiplied if they are conformable. If A is order k × n and B is of order n × 1 then the matrices are conformable. Using the partitioned matrix above, we have C = AB = a 1 · a 2 · . . . a k · b · 1 b · 2 · · · b · l = a 1 · b · 1 a 1 · b · 2 · · · a 1 · b · l a 2 · b · 1 a 2 · b · 2 · · · a 2 · b · l . . . . . . . . . . . . a k · b · 1 a k · b · 2 · · · a k · b · l (6) 5. Theorem 1.1 Let α and β be scalars and A , B , and C be matrices. Then when the operations involved are defined, the following properties hold a) A + B = B + A . b) ( A + B ) + C = A + ( B + C ) c) α ( A + B ) = αA + αB d) ( α + β ) = αA + βB e) A A = A + ( A ) = [0] f) A ( B + C ) = AC + BC 2
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVI Fall 2010 g) ( A + B ) C = AC + BC h) ( AB ) C = A ( BC ) 6. The transpose of an m × n matrix is a n × m matrix with the rows and columns interchanged. The transpose of A is denoted A . 7. Theorem 1.2 Let α and β be scalars and A and B be ma- trices. Then when defined, the following hold a) ( αA ) = αA b) ( A ) = A c) ( αA + βB ) = αA + βB d) ( AB ) = B A 8. The trace is a function defined as the sum of the diagonal elements of a square matrix tr ( A ) = m i =1 a ii (7) 9. Theorem 1.3 Let α be scalar and A and B be matrices. Then when the appropriate operations are defined, we have a) tr ( A ) = tr ( A ) b) tr ( αA ) = α tr ( A ) c) tr ( A + B ) = tr ( A ) + tr ( B ) d) tr ( AB ) = tr ( B A ) e) tr ( A A ) = 0 if and only if A = [0] 10. Traces can be very useful in statistical applications. For ex- ample, natural logarithm of the normal distribution function can be written as Λ n ( μ, Ω) = 1 2 mn ln (2 π ) 1 2 n ln ( | Ω | ) 1 2 tr Ω 1 Z Z = i = 1 n ( y i μ ) ( y i μ ) (8) 11. The Determinant is another function of square matrices. In
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