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Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig R

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Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig R

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    1. Fin500J Mathematical Foundations in Finance Topic 1: Matrix Algebra Philip H. Dybvig Reference: Mathematics for Economists, Carl Simon and Lawrence Blume, Chapter 8 Chapter 9 and Chapter 16 Slides designed by Yajun Wang 1 Fall 2010 Olin Business School Fin500J Topic 1

    2. Outline Definition of a Matrix Operations of Matrices Determinants Inverse of a Matrix Linear System Matrix Definiteness Fall 2010 Olin Business School 2 Fin500J Topic 1

    3. Matrix (Basic Definitions) 3 Fall 2010 Olin Business School Fin500J Topic 1

    4. Operations with Matrices (Sum, Difference) 4 Fall 2010 Olin Business School Fin500J Topic 1

    5. Operations with Matrices (Scalar Multiple) 5 Fall 2010 Olin Business School Fin500J Topic 1

    6. Operations with Matrices (Product) 6 Fall 2010 Olin Business School Fin500J Topic 1

    7. Laws of Matrix Algebra The matrix addition, subtraction, scalar multiplication and matrix multiplication, have the following properties. Fall 2010 Olin Business School 7 Fin500J Topic 1

    8. Operations with Matrices (Transpose) 8 Fall 2010 Olin Business School Fin500J Topic 1

    9. Determinants Determinant is a scalar Defined for a square matrix Is the sum of selected products of the elements of the matrix, each product being multiplied by +1 or -1 9 Fall 2010 Olin Business School Fin500J Topic 1

    10. Determinants The determinant of a 3 ×3 matrix is 10 The determinant of a 2 ×2 matrix A is Fall 2010 Olin Business School Fin500J Topic 1

    11. Inverse of a Matrix Definition. If A is a square matrix, i.e., A has dimensions n×n. Matrix A is nonsingular or invertible if there exists a matrix B such that AB=BA=In. For example. 11 Fall 2010 Olin Business School Fin500J Topic 1

    12. Calculation of Inversion using Determinants 12 Fall 2010 Olin Business School Fin500J Topic 1

    13. Calculation of Inversion using Determinants 13 Fall 2010 Olin Business School Fin500J Topic 1

    14. Calculation of Inversion using Gaussian Elimination 14 Fall 2010 Olin Business School Fin500J Topic 1

    15. Calculation of inversion using Gaussian elimination 15 Fall 2010 Olin Business School Fin500J Topic 1

    16. Example 16 Fall 2010 Olin Business School Fin500J Topic 1

    17. 17 Fall 2010 Olin Business School Systems of Equations in Matrix Form Fin500J Topic 1

    18. Example: solve the linear system 18 Fall 2010 Olin Business School Fin500J Topic 1

    19. 19 Fall 2010 Olin Business School Fin500J Topic 1

    20. 20 Fall 2010 Olin Business School Fin500J Topic 1

    21. Positive Definite Matrix Fall 2010 Olin Business School 21 Fin500J Topic 1

    22. Negative Definite, Positive Semidefinite, Negative Semidefinite, Indefinite Matrix Fall 2010 Olin Business School 22 Fin500J Topic 1

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