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Welcome to MAR 6658

Welcome to MAR 6658. Course Title Quantitative Methods in Marketing IV: Psychometric and Econometric Techniques Prerequisites MAR 6507 or instructor permission Instructor Charles Hofacker Meeting Tue 1:00-5:00 Contact Info Email : chofack @ cob.fsu.edu Office : RBB 255

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Welcome to MAR 6658

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  1. Welcome to MAR 6658 Course Title Quantitative Methods in Marketing IV: Psychometric and Econometric Techniques Prerequisites MAR 6507 or instructor permission Instructor Charles Hofacker Meeting Tue 1:00-5:00 Contact Info Email: chofack @ cob.fsu.edu Office: RBB 255 Hours: T/ R 11:00-12:00 Grades Two exams plus homework

  2. Ready to Get Going?

  3. Vectors and Transposing Vectors An m element column vector A q element row vector Transpose the row Transpose the column

  4. A Matrix Is A Set of Vectors • X is an n · m matrix • First subscript indexes rows • Second subscript indexes columns

  5. The Transpose of a Matrix Note that (X')' = X

  6. 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

  7. The Dot Subscript Reduction Operator – Columns Or we can keep track of each column of X: So that X is

  8. The Equals Sign A = B iff aij = bij for all i, j. The matrices must have the same order.

  9. Some Special Matrices Diagonal Scalar cI Unit 1

  10. More Special Matrices Null Symmetric Identity

  11. Matrix Addition • Adding two matrices means adding corresponding elements. • The two matrices must be conformable.

  12. Commutative: A + B = B + A Associative: A + (B + C) = (A + B) + C Identity: A + 0 = A Properties of Matrix Addition

  13. 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

  14. Orthogonal Vectors Two vectors x and y are said to be orthogonal if x =[2 1]

  15. Scalar Multiplication Associative: c1(c2A) = (c1c2)A Distributive: (c1 + c2) A = c1A + c2A

  16. Matrix Multiplication

  17. Partitioned Matrices Visually, matrices act like scalars And here is a little example

  18. The Cross Product Matrix B Keeping track of the columns of X

  19. The Cross Product Matrix 2 Keeping track of the rows of X

  20. 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

  21. 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.

  22. Solving a Linear System Consider the following system in two unknowns: The key to solving this is in the denominator below:

  23. An Inverse for Matrices ax = y a-1ax = a-1y 1x = a-1y x = a-1y Ax = y A-1Ax = A-1y Ix = A-1y x = A-1y • 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.

  24. The Inverse of a 2 · 2

  25. The Inverse of a Product Inverse of a Product: (AB)-1 = B-1A-1

  26. Quadratic Form (Bilinear form is where the pre- and post-multiplying vectors are not necessarily identical)

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