06d-MatrixAlgebraIntro

# 06d-MatrixAlgebraIntro - EXST7015 Matrix (Part 1) Matrix...

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EXST7015 Matrix Algebra Geaghan Matrix (Part 1) Introduction & Simple Linear Regression Page 1 06d-MatrixAlgebraIntro.doc A. MATRIX STRUCTURE AND NOTATION 1) A matrix is a rectangular arrangement of numbers. The matrix is usually denoted by a capital letter. A = 13 79 L N M O Q P D = 424 160 305 230 L N M M M M O Q P P P P 2) The dimensions of a matrix are given by the number of rows and columns in the matrix (i.e. the dimensions are r by c). For the matrices above, A is 2 by 2 D is 4 by 3 3) The individual elements of a matrix can be referred to by specifying the row and column in which it occurs. Lower case numbers are used to represent individual elements, and should match the upper case letter used to denote matrix. For example, individual elements from matrices A and D above can be referred to as, a 11 = 1 a 21 = 7 d 22 = 6 d 12 = 2 B. TYPES OF MATRICES 1) Square matrix - the number of rows and columns are equal. Matrix A above is a square matrix (2 by 2), matrix D is not (4 by 3). A symmetric matrix is an important variation of the square matrix. In a symmetric matrix, the value in position “ij" equals the value in position “ji" (where i j). For example, if c 31 = 5 then c 13 is also 5. 2) Scalar - a single number can be thought of as a 1 by 1 matrix and is called a scalar. 3) Vector - a single column or single row of numbers is called a vector. The dimensions of a row vector are (1 by c), where "c" is the number of columns, and the dimensions of a column vector (r by 1), where "r" is the number of rows. 4) Identity matrix - this special square matrix consists of all ones on the main diagonal, or principal diagonal, and zeros in all the off diagonal positions. The following are examples of identity matrices, E = 100 010 001 L N M M M O Q P P P F = 1000 0100 0010 0001 L N M M M M O Q P P P P The diagonal matrix is a generalization of the identity matrix. A diagonal matrix can have any value on the main diagonal, but also has zeros in the off diagonal positions.

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EXST7015 Matrix Algebra Geaghan Matrix (Part 1) Introduction & Simple Linear Regression Page 2 06d-MatrixAlgebraIntro.doc C. MATRIX TRANSPOSE The transpose of a matrix consists of a new matrix such that the rows of the original matrix become the columns of the transpose matrix. The transpose matrix is denoted with the same letter as the original matrix followed by a prime (e.g. the transpose of X is X). D = 424 160 305 230 L N M M M M O Q P P P P D = 4132 2603 4050 L N M M M O Q P P P D. MATRIX ADDITION AND SUBTRACTION Matrices to be added or subtracted must be of the same dimensions. Each element of the first matrix, (a) is added (or subtracted) from the corresponding element of the second matrix, (b). A = 12 34 90 L N M M M O Q P P P B = 14 44 L N M M M O Q P P P A+B = 11 2 4 31 44 94 04 + + ++ −+ L N M M M O Q P P P = 22 48 54 L N M M M O Q P P P E. MATRIX MULTIPLICATION Multiplication by a scalar - in this type of multiplication each element of the matrix is simply multiplied, element by element, by the scalar value.
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## This note was uploaded on 12/29/2011 for the course EXST 7087 taught by Professor Wang,j during the Fall '08 term at LSU.

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06d-MatrixAlgebraIntro - EXST7015 Matrix (Part 1) Matrix...

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