102c_lecture1

102c_lecture1 - MATRIX ALGEBRA Why is it useful? - simple...

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1 MATRIX ALGEBRA Why is it useful? - simple k-variables OLS expressions (and estimators other than OLS) - any data set you deal with is a big matrix - useful at the programming stage (i.e., for Gauss, Matlab – not for Stata) - it’s elegant! A matrix is a rectangular set of elements ordered in rows and columns,  RC R C C C R a a a a a a a a ... ... ... ... .... ... ... ... 1 2 22 21 1 12 11 A In this matrix there are R rows and C columns. If R=C the matrix is a square one, otherwise is a rectangular one. The diagonal of a square matrix (\) is called “main (or principal) diagonal” (only square matrices have a main diagonal). Sometimes this is indicated as diag ( A ). The (/) diagonal of a square matrix is called “opposite diagonal”. The generic element of the matrix is a ij . The first subscript (i) indexes the row where this element is located. The second subscript indexes the column (j) where this element is located. So a 25 is the element in the 2 nd row and 5 th column of A . The “elements” of the matrix can be everything – even abstract objects, letters, etc – but in econometrics they are numbers. A matrix is said to be symmetric if a ij = a ji . The dimension of a matrix is defined by the number of rows and columns it has. So a matrix with R rows and C columns has dimension (R C) (written under A – but can be omitted if not needed). The dimension of a matrix A is indicated as dim ( A ). A vector is a particular type of matrix with either one row (in which case is called a row vector) or one column (column vector). A vector (with no denomination) is conventionally assumed to be a column vector. Examples
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2    1 21 11 1 1 12 11 1 ... ... R R C C d d d b b b D B B is a row vector and D a column vector. A scalar is a particular type of matrix with just one row and one column, i.e., a singleton. Some (apparently silly) properties of matrices/remarks about matrices 1) Suppose you have two matrixes A and B with generic elements a ij and b ij . Then A = B iff a ij = b ij for all i,j . 2) Suppose you have two matrixes A and B with generic elements a ij and b ij . You can add them or subtract them ( B A ) iff they have the same dimension. 1 The resulting matrix C has a generic element ij ij ij b a c . Example. Define  2 6 5 8 4 2 5 6 2 1 , 1 4 0 7 , 3 2 5 1 B A B A C B A However, A+C or A-C are not defined (and neither are B+C or B-C). 3) A is always a feasible operation as long as is a scalar. The resulting matrix B = A will have generic element b ij = a ij .
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This note was uploaded on 07/28/2011 for the course ECON 102C taught by Professor Pistaferri,l during the Spring '11 term at Stanford.

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102c_lecture1 - MATRIX ALGEBRA Why is it useful? - simple...

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