matrix

Matrix - An Overview of Matrix Algebra x Definitions Operations x STATA matrix commands What is it x Matrix algebra is a means of making

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Unformatted text preview: An Overview of Matrix Algebra x Definitions Operations x STATA matrix commands What is it? x Matrix algebra is a means of making calculations upon arrays of numbers (or data). x Your data set is a matrix Why use it? x Matrix algebra makes mathematical expression and computation easier. x It allows you to get rid of cumbersome notation, concentrate on the concepts involved and understand where your results come from. Definitions - scalar x a scalar is a number – (denoted with regular type: 1 or 22) Definitions - vector x Vector: a single row or column of numbers – denoted with bold small letters – row vector a = [1 2 3 4 5] – column vector b= 1 2 3 4 5 Definitions - Matrix xA matrix is an array of numbers A a11 a12 a13 = a 21 a 22 a 23 xDenoted with a bold Capital letter xAll matrices have an order (or dimension): that is, the number of rows × the number of columns. So, A is 2 by 3 or (2 × 3). Definitions x A square matrix is a matrix that has the same number of rows and columns (n × n) Matrix Equality x Two matrices are equal if and only if – they both have the same number of rows and the same number of columns – their corresponding elements are equal Matrix Operations x Transposition x Addition and Subtraction x Multiplication x Inversion The Transpose of a Matrix: A' x The transpose of a matrix is a new matrix that is formed by interchanging the rows and columns. x The transpose of A is denoted by A' or (AT) Example of a transpose x Thus, a11 a12 A = a21 a22 a a 31 32 a11 a21 a31 A' = a12 a22 a32 x If A = A', then A is symmetric Addition and Subtraction x Two matrices may be added (or subtracted) iff they are the same order. x Simply add (or subtract) the corresponding elements. So, A + B = C yields Addition and Subtraction (cont.) a11 a12 b11 b12 c11 c12 a a22 + b21 b22 = c 21 c 22 21 a31 a32 b31 b32 c31 c32 xWhere a11 + b11 = c11 a12 + b12 = c12 a21 + b21 = c 21 a22 + b22 = c 22 a31 + b31 = c31 a32 + b32 = c32 Matrix Multiplication x To multiply a scalar times a matrix, simply multiply each element of the matrix by the scalar quantity a11 a12 ka11 ka12 k = ka21 ka22 a21 a22 Matrix Multiplication (cont.) x To multiply a matrix times a matrix, we write • AB (A times B) x This is pre-multiplying B by A, or post- multiplying A by B. Matrix Multiplication (cont.) x In order to multiply matrices, they must be CONFORMABLE x that is, the number of columns in A must equal the number of rows in B x So, A× B=C (m × n) × (n × p) = (m × p) Matrix Multiplication (cont.) x (m × n) × (p × n) = cannot be done x (1 × n) × (n × 1) = a scalar (1x1) Matrix Multiplication (cont.) x Thus x where a11 a12 a13 b11 b12 c11 c12 a a a23 x b21 b22 = c 21 c 22 21 22 a31 a32 a33 b31 b32 c31 c32 c11 = a11b11 + a12 b21 + a13b31 c12 = a11b12 + a12 b22 + a13b32 c 21 = a21b11 + a22 b21 + a23b31 c 22 = a21b12 + a22 b22 + a23b32 c31 = a31b11 + a32 b21 + a33b31 c32 = a31b12 + a32 b22 + a33b32 Matrix Multiplication- an example x Thus 1 4 7 1 4 c11 c12 30 66 2 5 8 x 2 5 = c c 22 = 36 81 21 3 6 9 3 6 c31 c32 42 96 x where c11 = 1 * 1 + 4 * 2 + 7 * 3 = 30 c12 = 1 * 4 + 4 * 5 + 7 * 6 = 66 c 21 = 2 * 1 + 5 * 2 + 8 * 3 = 36 c 22 = 2 * 4 + 5 * 5 + 8 * 6 = 81 c31 = 3 * 1 + 6 * 2 + 9 * 3 = 42 c32 = 3 * 4 + 6 * 5 + 9 * 6 = 96 Properties x AB does not necessarily equal BA x (BA may even be an impossible operation) x For example, A × (2 × 3) × B × (3 × 2) × B = (3 × 2) = A = (2 × 3) = C (2 × 2) D (3 × 3) Properties x Matrix multiplication is Associative A(BC) = (AB)C x Multiplication and transposition (AB)' = B'A' A popular matrix: X'X 1 1 X= 1 X' X = x11 x12 x1n 1 1 1 × x11 x12 x1n 1 x11 1 x12 1 x 1n = n n x ∑ 1i i =1 ∑ x1i i =1 n 2 ∑ x1i i =1 n Another popular matrix: e'e e = e' e e1 e 2 en = [ e1 e2 en ] e1 e × 2 en = n ei2 ∑ i =1 Special matrices x There are a number of special matrices – Diagonal – Null – Identity Diagonal Matrices – A diagonal matrix is a square matrix that has values on the diagonal with all off-diagonal entities being zero. a11 0 0 a 22 0 0 0 0 0 0 a33 0 0 0 0 a44 Identity Matrix x An identity matrix is a diagonal matrix where the diagonal elements all equal one. I= 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 A× I=A Null Matrix x A square matrix where all elements equal zero. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The Determinant of a Matrix x The determinant of a matrix A is denoted by |A| (or det(A)). x Determinants exist only for square matrices. x They are a matrix characteristic, and they are also difficult to compute The Determinant for a 2x2 matrix x If A = a11 a12 a 21 a22 x Then A = a11a22 − a12a21 Properties of Determinates x Determinants have several mathematical properties which are useful in matrix manipulations. – 1 |A|=|A'|. – 2. If a row or column of A = 0, then |A|= 0. – 3. If every value in a row or column 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 or columns are identical, |A| = 0. – 6. If two rows or columns are linear combination of each other, |A| = 0 Properties of Determinants – 7. |A| remains unchanged if each element of a row or each element multiplied by a constant, is added to any other row. – 8. |AB| = |A| |B| – 9. Det of a diagonal matrix = product of the diagonal elements Rank x The rank of a matrix is defined as x rank(A) = number of linearly independent rows = the number of linearly independent columns. x A set of vectors is said to be linearly dependent if scalars c1, c2, …, cn (not all zero) can be found such that c1a1 + c2a2 + … + cnan = 0 For example, a = [1 21 12] and b = [1/3 7 4] are linearly dependent x A matrix A of dimension n × p (p < n) is of rank p. Then A has maximum possible rank and is said to be of full rank. x In general, the maximum possible rank of an n × p matrix A is min(n,p). The Inverse of a Matrix (A-1) x For an n × n matrix A, there may be a B such that AB = I = BA. x The inverse is analogous to a reciprocal x A matrix which has an inverse is nonsingular. x A matrix which does not have an inverse is singular. x An inverse exists only if A ≠0 Properties of inverse matrices x ( AB ) x ( A' ) x −1 −1 (A ) -1 −1 = BA -1 -1 = (A ) ' = A -1 How to find inverse matrixes? determinants? and more? a b x If A = c d A -1 = and |A| ≠ 0 1 det( A) d − b − c a x Otherwise, we use STATA ...
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This note was uploaded on 10/21/2011 for the course PS 233 taught by Professor Staff during the Spring '11 term at Duke.

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