MathematicalMarketing Slide 1.1 Linear Algebra Welcome to MAR 6658 Course Title Quantitative Methods in Marketing IV: Psychometric and Econometric Techniques PrerequisitesMAR 6507 or instructor permission InstructorCharles Hofacker MeetingTue 1:00-5:00 Contact Info cob.fsu.edu Office: RBB 255 Hours: T/ R 11:00-12:00 GradesTwo exams plus homework

MathematicalMarketing Slide 1.2 Linear Algebra Ready to Get Going?

MathematicalMarketing Slide 1.3 Linear Algebra Vectors and Transposing Vectors An m element column vectorA q element row vector Transpose the column Transpose the row

MathematicalMarketing Slide 1.4 Linear Algebra A Matrix Is A Set of Vectors X is an n · m matrix First subscript indexes rows Second subscript indexes columns

MathematicalMarketing Slide 1.5 Linear Algebra The Transpose of a Matrix Note that (X')' = X

MathematicalMarketing Slide 1.6 Linear Algebra The Dot Subscript Reduction Operator - Rows We can display an intermediate amount of detail by separately keeping track of each row: So the matrix X becomes

MathematicalMarketing Slide 1.7 Linear Algebra The Dot Subscript Reduction Operator – Columns Or we can keep track of each column of X: So that X is

MathematicalMarketing Slide 1.8 Linear Algebra The Equals Sign A = B iff a ij = b ij for all i, j. The matrices must have the same order.

MathematicalMarketing Slide 1.9 Linear Algebra Some Special Matrices Diagonal Scalar cI Unit 1

MathematicalMarketing Slide 1.10 Linear Algebra More Special Matrices Null Symmetric Identity

MathematicalMarketing Slide 1.11 Linear Algebra Matrix Addition  Adding two matrices means adding corresponding elements.  The two matrices must be conformable.

MathematicalMarketing Slide 1.12 Linear Algebra Properties of Matrix Addition  Commutative:A + B = B + A  Associative:A + (B + C) = (A + B) + C  Identity:A + 0 = A

MathematicalMarketing Slide 1.13 Linear Algebra Vector Multiplication Vector multiplication works with a row on the left and a column on the right. There are a lot of names for this: linear combination dot product scalar product inner product

MathematicalMarketing Slide 1.14 Linear Algebra Orthogonal Vectors x =[2 1] Two vectors x and y are said to be orthogonal if

MathematicalMarketing Slide 1.15 Linear Algebra Scalar Multiplication Associative:c 1 (c 2 A) = (c 1 c 2 )A Distributive:(c 1 + c 2 ) A = c 1 A + c 2 A

MathematicalMarketing Slide 1.16 Linear Algebra Matrix Multiplication

MathematicalMarketing Slide 1.17 Linear Algebra Partitioned Matrices Visually, matrices act like scalars And here is a little example

MathematicalMarketing Slide 1.18 Linear Algebra The Cross Product Matrix B Keeping track of the columns of X

MathematicalMarketing Slide 1.19 Linear Algebra The Cross Product Matrix 2 Keeping track of the rows of X

MathematicalMarketing Slide 1.20 Linear Algebra Properties of Multiplication Scalar Multiplication:  Commutative:cA = Ac  Associative: A(cB) = (cA)B = c(AB) Matrix Multiplication:  Associative:(AB)C = A(BC)  Right Distributive: A[B + C] = AB + AC  Left Distributive:[B + C]A = BA + CA  Transpose of a Product(BA)' = A'B'  IdentityIA = AI = A

MathematicalMarketing Slide 1.21 Linear Algebra The Trace of a Matrix Tr[AB] = Tr[BA]. The theorem is applicable if both A and B are square, or if A is m · n and B is n · m Note that for a scalar s, Tr s = s.

MathematicalMarketing Slide 1.22 Linear Algebra Solving a Linear System Consider the following system in two unknowns: The key to solving this is in the denominator below:

MathematicalMarketing Slide 1.23 Linear Algebra An Inverse for Matrices ax = y a -1 ax = a -1 y 1x = a -1 y x = a -1 y Ax = y A -1 Ax = A -1 y Ix = A -1 y x = A -1 y Scalars: One Equation and One Unknown Matrices: N Equations and N Unkowns We just need to find a matrix A -1 such that AA -1 = I.

MathematicalMarketing Slide 1.24 Linear Algebra The Inverse of a 2 · 2

MathematicalMarketing Slide 1.25 Linear Algebra The Inverse of a Product Inverse of a Product:(AB) -1 = B -1 A -1

MathematicalMarketing Slide 1.26 Linear Algebra Quadratic Form (Bilinear form is where the pre- and post-multiplying vectors are not necessarily identical)