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

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
Review of Matrix Algebra and Vector Spaces Charles B. Moss August 3, 2010 I. Review of Elementary Matrix Algebra A. Basic DeFnitions 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 = h a · 1 a · 2 a · n i 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 · = h a 11 a 12 a 1 n i a 2 · = h a 21 a 22 a 2 n i . . . a m · = h a m 1 a m 2 a mn i (3) 2. The sum of two identically dimensioned matrices can be ex- pressed as 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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 ±rst 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 · h b · 1 b · 2 ··· b · l i = 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 de±ned, 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
Background image of page 2
AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXVI Fall 2010 g) ( A + B ) C = AC + BC h) ( AB ) C = A ( ) 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 0 . 7. Theorem 1.2 Let α and β be scalars and A and B be ma- trices. Then when de±ned, the following hold a) ( αA ) 0 = αA 0 b) ( A 0 ) 0 = A c) ( αA + βB ) 0 = αA 0 + 0 d) ( AB ) 0 = B 0 A 0 8. The trace is a function de±ned 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 de±ned, we have a) tr ( A 0 )=tr( A ) b) tr ( αA α tr ( A ) c) tr ( A + B A )+tr( B ) d) tr ( AB B 0 A 0 ) e) tr ( A 0 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 ) 0 (8) 11. The Determinant is another function of square matrices. In
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

Page1 / 11

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online