Lecture 26-2007 - Review of Matrix Algebra and Vector...

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Review of Matrix Algebra and Vector Spaces: Lecture XXVI I. Review of Elementary Matrix Algebra A. The material for this lecture is found in James R. Schott Matrix Analysis for Statistics (New York: John Wiley & Sons, Inc. 1997). B. Basic definitions 1. A matrix A of size m n is an m n rectangular array of scalars: 11 12 1 21 22 2 1 2 n n m m mn a a a a a a A a a a It is sometimes useful to partition matrices into vectors. 11 12 1 21 22 2 •1 •2 1 2 1• 2• 3• n n n m m mn a a a a a a A a a a a a a a a a where 1 2 1 2 j j j i i i im mj a a a or a a a a a 2. The sum of two identically dimensioned matrices can be expressed as ij ij A B a b
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XXVI Professor Charles Moss Fall 2007 2 3. In order to multiply a matrix by a scalar, multiply each element of the matrix by the scalar. 4. In order to discuss matrix multiplication, we first discuss vector 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 1 n , then the vectors are conformable and the multiplication becomes: 1 n i i i z x y x y 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 1 n . then the matrices are conformable. Using the partitioned matrix above, we have 1• 2• •1 •2 1• •1 1• •2 1• 2• •1 2• •2 2• •1 •2 l k l l k k k l a a C AB b b b a a b a b a b a b a b a b a b a b a b 5. Theorem 1.1 Let and be scalars and A , B , and C be matrices. Then when the operations involved are defined, the following properties hold:
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture XXVI Professor Charles Moss Fall 2007 3 a) A B B A . b) A B C A B C . c) A B A B . d) A A A . e) 0 A A A A . f) A B C AB AC . 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 and be scalars and A and B be matrices. Then when defined, the following hold a) A A . b) A A . c) A B A B . d) AB B A . 8. The trace is a function defined as the sum of the diagonal elements of a square matrix.
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