Introduction to Financial Modeling MGT 4850 Spring 2008 University of Lethbridge

Topics The power of Numbers Quantitative Finance Risk and Return Asset Pricing Risk Management and Hedging Volatility Models Matrix Algebra

MATRIX ALGEBRA Definition –Row vector –Column vector

Matrix Addition and Scalar Multiplication Definition: Two matrices A = [a ij ] and B = [b ij ] are said to be equal if Equality of these matrices have the same size, and for each index pair (i, j), a ij = b ij, Matrices that is, corresponding entries of A and B are equal.

Matrix Addition and Subtraction Let A = [a ij ] and B = [b ij ] be m × n matrices. Then the sum of the matrices, denoted by A + B, is the m × n matrix defined by the formula A + B = [a ij + b ij ]. The negative of the matrix A, denoted by −A, is defined by the formula −A = [−a ij ]. The difference of A and B, denoted by A−B, is defined by the formula A − B = [a ij − b ij ].

Scalar Multiplication Let A = [a ij ] be an m × n matrix and c a scalar. Then the product of the scalar c with the matrix A, denoted by cA, is defined by the formula Scalar cA = [ca ij ].

Linear Combinations A linear combination of the matrices A 1,A 2,..., A n is an expression of the form c 1 A 1 + c 2 A 2 + ・ ・ ・ + c n A n

Laws of Arithmetic Let A,B,C be matrices of the same size m × n, 0 the m × n zero matrix, and c and d scalars. (1) (Closure Law) A + B is an m × n matrix. (2) (Associative Law) (A + B) + C = A + (B + C) (3) (Commutative Law) A + B = B + A (4) (Identity Law) A + 0 = A (5) (Inverse Law) A + (−A) = 0 (6) (Closure Law) cA is an m × n matrix.

Laws of Arithmetic (II) (7) (Associative Law) c(dA) = (cd)A (8) (Distributive Law) (c + d)A = cA + dA (9) (Distributive Law) c(A + B) = cA + cB (10) (Monoidal Law) 1A = A

Matrix Multiplication