Presentation on theme: "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."— Presentation transcript:
1 Matrix AlgebraMatrix algebra is a means of expressing large numbers of calculations made upon ordered sets of numbers.Often referred to as Linear AlgebraMany 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.
2 Definitions - scalar scalar - a number denoted with regular type as is scalar algebra or [a]
3 Definitions - vector vector - a single row or column of numbers denoted with bold small lettersrow vector a =column vector x =
4 Definitions - Matrix A matrix is a set of rows and columns of numbers Denoted with a bold Capital letterAll matrices (and vectors) have an order - that is the number of rows x the number of columns.Thus A =
5 Matrix EqualityTwo matrices are equal iff (if and only if) all of their elements are identicalNote: your data set is a matrix.
6 Matrix Operations Addition and Subtraction Multiplication TranspositionInversion
7 Addition and Subtraction Two matrices may be added iff they are the same order.Simply add the corresponding elements
9 Scalar Multiplication To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity
10 Matrix Multiplication (cont.) To multiply a matrix times a matrix, we writeA times B as ABThis is pre-multiplying B by A, or post-multiplying A by B.
11 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 donea (1xn) x (nx1) = a scalar (1x1)
12 Matrix Multiplication (cont.) The general rule for matrix multiplication is:
13 Matrix multiplication is not Commutative AB does not necessarily equal BA(BA may even be an impossible operation)
14 Yet matrix multiplication is Associative A(BC) = (AB)C
15 Special matrices There are a number of special matrices Square DiagonalSymmetricNullIdentity
16 Square matrixA square matrix is just what it sounds like, an nxn matrixSquare matrices are quite useful for describing the properties or interrelationships among a set of things – like the correlation matric for your dataset.
17 Diagonal MatricesA diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero.
18 Symmetric MatrixAll of the elements in the upper right portion of the matrix are identical to those in the lower left.For example, the correlation matrix
19 Identity MatrixThe identity matrix I 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.
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)Idempotent means applying the same operation gives invariant results – e.g. 1 x 1 = 1 (see further…)The transpose of a sum = sum of transposes(A + B + C)’ = A’ + B’ + C’The transpose of a product = the product of the transposes in reverse order(ABC)’ = C’B’A’
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 - contSince 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 =ThenThat one is easy
29 DeterminantsFor 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.)
32 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.
33 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 |AB| = |A| |B|8. Det of a diagonal matrix = product of the diagonal elements
34 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
35 Inverse by Row or column operations Set up a tableau matrixA tableau for inversions consists of the matrix to be inverted post multiplied by a conformable identity matrix.
36 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 IWhen you are done; A = I & I = A-1
37 The Tableau Method of Matrix Inversion: An Example Step 1: Set up TableauStep 2: Add –2(Row 1) to Row 2
38 Matrix Inversion – cont. Step 3: Add –1(Row 1) to Row 3Step 4: Multiply Row 2 by –1/3
43 Using Stata to do matrix Algebra Try the following command setmatrix input A= (1, 2 \ 3, 4)matrix list Amatrix input B= (5, 6 \ 7, 8)matrix list Bmatrix define C = A + Bmatrix list Cmatrix X = (1, 4, 3 \ 2, 5, 4 \ 1, -3, -2)matrix Xinv = inv(X)matrix list Xinv
44 The Matrix ModelThe 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
45 The Matrix Model (cont.) The matrices in are represented byNote that we postmultiply X by B since this order makes them conformable.
46 The Assumptions of the Model Scalar Version 1. The ei's are normally distributed.2. E(ei) = 03. E(ei2) = 24. E(eiej) = 0 (ij)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.
47 The Assumptions of the Model: The Matrix Version These same assumptions expressed in matrix format are:1. e N(0,)2. = 2I3. The elements of X are fixed in repeated samples and (1/ n)X'X is nonsingular and its elements are finite
48 Derivation of B's in matrix notation Given the matrix algebra model we can replicate the least squares normal equations in matrix format.We need to minimize ee’ which is the sum of squared errorsSetting the derivative equal to 0 we ultimately getNote that X’X is called the sums-of-squares and cross-products matrix.