11%20Matrix%20Notation%202_19_08

11%20Matrix%20Notation%202_19_08 - Matrix Introduction to...

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1 1 Introduction to Matrix Algebra Useful for Statistics Peng Liu 2/19/2008 2 Matrix ± A matrix is a rectangular array of elements arranged in rows and columns. An example: [] ij a a a a a a a a a A = = = 24 14 23 13 22 12 21 11 8 4 7 3 6 2 5 1 Column 1 2 3 4 Row 1 Row 2 Row index i=1,2 and Column index j = 1,2,3 3 Vector ± A matrix containing only one column is called a (column) vector. ± The transpose of a vector a is the corresponding row array and is denoted as a ’. [ ] 3 2 1 3 2 1 3 2 1 ' 3 2 1 a a a a a a a a = = = = 4 Matrix dimension ± Matrix dimension of r x k : r rows and k columns ± a ij = the element of row i and column j ± Matrix A = [a ij ] ± Note that a (column) vector is a matrix with column dimension 1. The dimension is often called the length for a vector.
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2 5 Transpose of matrix ± Transpose of matrix A is denoted as A’ (or, t(A)) and is obtained by interchanging corresponding rows and columns of A. ± If A has dimension r x k , then its transpose A’ has dimension k x r. = = 8 4 7 3 6 2 5 1 ' 8 4 7 3 6 2 5 1 A A 6 Matrix addition and subtraction ± Adding or subtracting two matrices requires that they have the same dimension. ± In general: ] [ - and ] [ ] [ and ] [ ij ij ij ij ij c r ij c r b a B A b a B A b B a A = + = + = = × × 7 Matrix addition and subtraction: example = = + = = 6 0 5 0 4 0 3 0 10 8 9 6 8 4 7 2 2 4 2 3 2 2 2 1 8 4 7 3 6 2 5 1 B A B A B A Dimension: 2 x 4 2 x 4 2 x 4 2 x 4 8 Vector multiplication: inner product ± Inner product of a row and a column vector of the same length: n n n n b a b a b a b b b a a a b a + + + = = ... : ) ,.., , ( ' 2 2 1 1 2 1 2 1 = 6 5 4 ) 3 , 2 , 1 (
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3 9 Matrix multiplication ± Not any pair of matrices can be multiplied together. ± The dimension of the two matrices should match to be multiplied. ± A is r 1 x c 1 . B is r 2 x c 2 . We can perform the multiplication AB if c 1 = r 2 .
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11%20Matrix%20Notation%202_19_08 - Matrix Introduction to...

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