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Unformatted text preview: Matrix • A matrix is a rectangular array of elements arranged in rows and columns • Dimension of a matrix is r x c – r = c square matrix – r = 1 (row) vector – 1 4 2 5 3 6 = A Matrix (cont) • elements are either numbers of symbols – each element is indicated by aij – rows are indicated by an i subscript – columns are indicated by a j subscript • each element is designated by the row and column that it is in 11 12 21 22 ij 31 32 1 4 a a 2 5 a a a ,i 1 3,j 1 2 3 6 a a = = = = = A Matrix Equality • two matrices are said to be equal if they have the same dimensions and all of the corresponding elements are equal Equality: Example Are the following matrices equal? a) b) 1 4 1 2 3 2 5 4 5 6 3 6 = = A B 1 4 1 4 2 5 3 5 3 6 3 6 = = A B Matrix Operations  Transpose • A transposition is performed by switching the rows and columns, indicated by a ‘ – A’ = B if aij = bji • Note: if A is r x c then B will be c x r – The transpose of a column vector is a 1 4 1 2 3 2 5 ' 4 5 6 3 6 = ⇒ = A A Matrix Operations – Addition/Subtraction • Two add or subtract two matrices, they must have the same dimensions • The addition or subtraction is done on an element by element bases 1 4 10 40 2 5 , 20 50 3 6 30 60 = = A B 1 10 4 40 11 44 2 20 5 50 22 55 3 30 6 60 33 66 + + = + + = + + A +B Matrix Operations  Multiplication • by a scalar (number) – every element of the matrix is multiplied by that number – In addition, a matrix can be factored 2 4 2k 4k ,k 3 5 3k 5k = = A A 2 4 2 4 1 k k k 3 5 3 5 k k = Matrix Operations  Multiplication • by another matrix – C = A B • • columns of A must equal rows of B • Resulting matrix has dimension rows of A x columns of B ij ik kj k c a b = ∑ 1 2 4 2 1 4 8 2 8 2 3 14 13 4 1 1 2 1 1 8 2 2 2 3 11 7 2 3 + + + + = = = + + + + AB Special Types of Matrices: Symmetric • if A = A’ , then A is said to be symmetric – a symmetric matrix has to be a square matrix 1 2 3 2 4 5 3 5 6 = A Special Types of Matrices:...
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This note was uploaded on 02/20/2012 for the course STAT 512 taught by Professor Staff during the Spring '08 term at Purdue.
 Spring '08
 Staff

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