1. REGRESSION (CONTINUED) Matrices & Matrix Algebra; Multivariate Regression LECTURE 5 Supplementary Readings: Wilks, chapters 6; Bevington, P.R., Robinson, D.K., Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, 1992.

2. Tutorial on Matrices and Matrix Algebra

3. VECTORS c is an N-length column vector b is an M-length row vector b T is an N-length column vector

4. VECTORS Can add two N-length row vectors or two N-length column vectors

5. VECTORS Can subtract two N-length row vectors or two N-length column vectors

6. VECTORS Can multiply an N-length vector by a constant

7. VECTORS Can multiply an K-length row vector by an K-length column vector ‘DOT PRODUCT’ or ‘INNER PRODUCT’ ‘EUCLIDEAN NORM’

8. VECTORS ‘DOT PRODUCT’ or ‘INNER PRODUCT’ ‘EUCLIDEAN NORM’ Note the close relationship with the linear correlation between two series

9. VECTORS Can multiply an N-length column vector by an M-length row vector ‘OUTER PRODUCT’

10. VECTORS Yields an NxM Matrix ‘OUTER PRODUCT’

11. MATRICES NxM Matrix

12. TRANSPOSE OF MATRIX NxM Matrix MxN Matrix

13. Can add two NxM Matrices MATRICES

14. Can multiply an NxK and KxM Matrix MATRICES

15. RULES FOR MATRIX ARITHMETIC Associative Laws Commutative Laws

16. NxN Matrix N IDENTITY MATRIX

18. DIAGONAL MATRICES

19. INVERSE OF A (SQUARE) MATRIX

20. INVERSE OF A MATRIX

22. Special Cases: 2x2 matrix If det(A)  0 then the matrix is “Invertible” Equivalent to the Matrix being of “full rank” (ie, there are no redundant rows in the matrix)

23. INVERSE OF A MATRIX Special Cases: diagonal matrix

24. UNITARY MATRIX Example: 2D Rotation Matrix Note that the inverse represents a rotation in the opposite direction

25. SOLUTION OF MATRIX EQUATION If A is invertible, We can write

26. Recall Linear Regression We can write this as a matrix equation,