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Matrix Algebra

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1. Matrix Algebra • Matrix algebra is a means of expressing large numbers of calculations made upon ordered sets of numbers. • Often referred to as Linear Algebra

2. Why use it? • Matrix algebra is used primarily to facilitate mathematical expression. • Many equations would be completely intractable if scalar mathematics had to be used. It is also important to note that the scalar algebra is under there somewhere.

3. Definitions - scalar • scalar - a number • denoted with regular type as is scalar algebra •  or [a]

4. Definitions - vector • vector - a single row or column of numbers • denoted with bold small letters • row vector • column vector

5. Definitions - Matrix • A matrix is a set of rows and columns of numbers • Denoted with a bold Capital letter • All matrices (and vectors) have an order - that is the number of rows x the number of columns. Thus A is

6. Matrix Equality • Thus two matrices are equal iff (if and only if) all of their elements are identical • Note: your data set is a matrix.

7. Matrix Operations • Addition and Subtraction • Multiplication • Transposition • Inversion

8. Addition and Subtraction • Two matrices may be added iff they are the same order. • Simply add the corresponding elements

9. Addition and Subtraction (cont.) • Where • Hence

10. Matrix Multiplication • To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity

11. Matrix Multiplication (cont.) • To multiply a matrix times a matrix, we write • A times B as AB • This is pre-multiplying B by A, or post-multiplying A by B.

12. Matrix Multiplication (cont.) • In order to multiply matrices, they must be conformable (the number of columns in A must equal the number of rows in B.) • an (mxn) x (nxp) = (mxp) • an (mxn) x (pxn) = cannot be done • a (1xn) x (nx1) = a scalar (1x1)

13. Matrix Multiplication (cont.) • Thus • where

14. Matrix Multiplication- an example • Thus • where

15. Matrix multiplication is not Commutative • AB does not necessarily equal BA • (BA may even be an impossible operation)

16. Yet matrix multiplication is Associative • A(BC) = (AB)C

17. Special matrices • There are a number of special matrices • Diagonal • Null • Identity

18. Diagonal Matrices • A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero.

19. Identity Matrix • An identity matrix is a diagonal matrix where the diagonal elements all equal one. It is used in a fashion analogous to multiplying through by "1" in scalar math.

20. Null Matrix • A square matrix where all elements equal zero. • Not usually ‘used’ so much as sometimes the result of a calculation. • Analogous to “a+b=0”

21. The Transpose of a Matrix A' • Taking the transpose is an operation that creates a new matrix based on an existing one. • The rows of A = the columns of A' • Hold upper left and lower right corners and rotate 180 degrees.

22. Example of a transpose

23. The Transpose of a Matrix A' • If A = A', then A is symmetric (i.e. correlation matrix) • If AA’ = A then A' is idempotent (and A' = A) • The transpose of a sum = sum of transposes • The transpose of a product = the product of the transposes in reverse order

24. An example: • Suppose that you wish to obtain the sum of squared errors from the vector e. Simply pre-multiply e by its transpose e'. • which, in matrices looks like

25. An example - cont • Since the matrix product is a scalar found by summing the elements of the vector squared.

26. The Determinant of a Matrix • The determinant of a matrix A is denoted by |A|. • Determinants exist only for square matrices. • They are a matrix characteristic, and they are also difficult to compute

27. The Determinant for a 2x2 matrix • If A = • Then • That one is easy

28. The Determinant for a 3x3 matrix • If A = • Then

29. Determinants • For 4 x 4 and up don't try. For those interested, expansion by minors and cofactors is the preferred method. • (However the spaghetti method works well! Simply duplicate all but the last column of the matrix next to the original and sum the products of the diagonals along the following pattern.)

30. Spaghetti Method of |A|

31. Properties of Determinates • Determinants have several mathematical properties which are useful in matrix manipulations. • 1 |A|=|A'|. • 2. If a row of A = 0, then |A|= 0. • 3. If every value in a row is multiplied by k, then |A| = k|A|. • 4. If two rows (or columns) are interchanged the sign, but not value, of |A| changes. • 5. If two rows are identical, |A| = 0.

32. Properties of Determinates • 6. |A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row. • 7. Det of product = product of Det's |A| = |A| |B| • 8. Det of a diagonal matrix = product of the diagonal elements

33. The Inverse of a Matrix (A-1) • For an nxn matrix A, there may be a B such that AB = I = BA. • The inverse is analogous to a reciprocal) • A matrix which has an inverse is nonsingular. • A matrix which does not have an inverse is singular. • An inverse exists only if

34. Inverse by Row or column operations • Set up a tableau matrix • A tableau for inversions consists of the matrix to be inverted post multiplied by a conformable identity matrix.

35. Matrix Inversion by Tableau Method • Rules: • You may interchange rows. • You may multiply a row by a scalar. • You may replace a row with the sum of that row and another row multiplied by a scalar. • Every operation performed on A must be performed on I • When you are done; A = I& I = A-1

36. The Tableau Method of Matrix Inversion: An Example • Step 1: Set up Tableau

37. Matrix Inversion – cont. • Step 2: Add –2(Row 1) to Row 2 • Step 3: Add –1(Row 1) to Row 3

38. Matrix Inversion – cont. • Step 4: Multiply Row 2 by –1/3 • Step 5: Add –4 (Row 2) to Row 1

39. Matrix Inversion – cont. • Step 6: Add 7(Row 2) to Row 3 • Step 7: Add Row 3 to Row 1

40. Matrix Inversion – cont. • Step 9: Add 2(Row 3) to Row 2 • Step 9: Multiply Row 3 by -3

41. Checking the calculation • Remember AA-1=I • Thus

42. The Matrix Model • The multiple regression model may be easily represented in matrix terms. • Where the Y, X, B and e are all matrices of data, coefficients, or residuals

43. The Matrix Model (cont.) • The matrices in are represented by • Note that we postmultiply X by B since this order makes them conformable.

44. The Assumptions of the ModelScalar Version • 1. The ei's are normally distributed. • 2. E(ei) = 0 • 3. E(ei2) = 2 • 4. E(eiej) = 0 (ij) • 5. X's are nonstochastic with values fixed in repeated samples and (Xik-Xbark)2/n is a finite nonzero number. • 6. The number of observations is greater than the number of coefficients estimated. • 7. No exact linear relationship exists between any of the explanatory variables.

45. The Assumptions of the Model: The Matrix Version • These same assumptions expressed in matrix format are: • 1. e  N(0,) • 2.  = 2I • 3. The elements of X are fixed in repeated samples and (1/ n)X'X is nonsingular and its elements are finite