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Unformatted text preview: Chapter 5. Matrix Algebra A Prelude to Multiple Regression Matrices are rectangular arrays of numbers and are denoted using boldface (mostly capital) symbols. Example: a 2 × 2 matrix (always #rows × #columns) A = bracketleftbigg 2 3 0 1 bracketrightbigg Example: a 4 × 2 matrix B , and a 2 × 3 matrix C B = 4 6 1 10 5 7 12 2 , C = bracketleftbigg 1 1 4 2 4 3 bracketrightbigg In general, an r × c matrix is given by A r × c = a 11 a 12 ··· a 1 j ··· a 1 c a 21 a 22 ··· a 2 j ··· a 2 c . . . . . . . . . . . . . . . . . . a i 1 a i 2 ··· a ij ··· a ic . . . . . . . . . . . . . . . . . . a r 1 a r 2 ··· a rj ··· a rc or in abbreviated form A r × c = [ a ij ] , i = 1 , 2 ,...,r, j = 1 , 2 ,...,c 1st subscript gives row#, 2nd subscript gives column# Where is a 79 or a 44 ? A matrix A is called square , if it has the same # of rows and columns ( r = c ). Example: A 2 × 2 = bracketleftbigg 2 . 7 7 . 1 . 4 3 . 4 bracketrightbigg Matrices having either 1 row ( r = 1) or 1 column ( c = 1) are called vectors . Example: Column vector A ( c = 1) and row vector C ′ ( r = 1) A = 4 7 13 , C ′ = bracketleftbig c 1 c 2 c 3 c 4 bracketrightbig Row vectors always have the prime! Transpose: A ′ is the transpose of A where A = 3 1 5 6 2 4 3 7 10 0 1 2 , A ′ = 3 2 10 1 4 5 3 1 6 7 2 A ′ is obtained by interchanging columns & rows of A a ij is the typical element of A a ′ ij is the typical element of A ′ a ij = a ′ ji ( a 12 = a ′ 21 ) Equality of Matrices: Two matrices A and B are said to be equal if they are of the same dimension and all corresponding elements are equal. A r × c = B r × c means a ij = b ij , i = 1 ,...,r , j = 1 ,...,c . Addition and Subtraction: To add or subtract matrices they must be of the same dimension. The result is another matrix of this dimension. If A 3 × 2 = 4 6 1 10 5 7 , B 3 × 2 = 2 3 0 1 7 5 , then the sum and difference are calculated elementwise: C = A + B = 4 + 2 6 + 3 1 + 0 10 + 1 5 + 7 7 + 5 = 6 9 1 11 12 12 D = A − B = 4 − 2 6 − 3 1 − 0 10 − 1 5 − 7 7 − 5 = 2 3 1 9 − 2 2 . Regression Analysis Remember, we had ( X 1 ,Y 1 ) , ( X 2 ,Y 2 ) ,..., ( X n ,Y n ) and wrote the SLR as Y i = E ( Y i ) + ǫ i , i = 1 , 2 ,...,n. Now we are able to write the above model as Y n × 1 = E ( Y n × 1 ) + ǫ n × 1 with the n × 1 column vectors Y = Y 1 Y 2 . . . Y n , E ( Y ) = E ( Y 1 ) E ( Y 2 ) ....
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This note was uploaded on 10/18/2011 for the course STA 4210 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
 Staff

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