Week 09 - QMA 2006S2 (Matrices Part 1)

Week 09 - QMA 2006S2 (Matrices Part 1) - Jessica Chung&...

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Unformatted text preview: Jessica Chung & Derek Hui Week 09 QMA PASS | 2006 S2 | Tues, 3-4pm | QUAD G042 Questions? Email us: [email protected] , [email protected] 1/5 MATRICES – SOME DEFINITIONS Matrix 11 12 13 21 22 23 31 32 33 2 7 8 5 1 9 4 3 2 a a a A a a a a a a & ¡ ¢ £ ¤ ¥ ¦ § = = ¤ ¥ ¦ § ¤ ¥ ¦ § ¨ © ª « & An array of numbers arranged into rows and columns & The order of a matrix is the number of rows multiplied by the number of columns ( m x n) & A matrix element is identified by its row and column number (e.g. a 32 = 3). Note that a ij & a ji & A square matrix is one with an equal number of rows and columns & The main (principal) diagonal stretches from top left to bottom right in a square matrix & Matrix equality exists if, and only if, the order and all elements of the matrices are the same Row and Column Vectors [ ] 5 8 2 A = 7 3 1 A ¢ £ ¦ § = ¦ § ¦ § ¨ © & A row vector has only a single row (is horizontal) & A column vector has only a single column (is vertical) Diagonal and Identity Matrices 5 2 7 A ¢ £ ¦ § = ¦ § ¦ § ¨ © 1 1 1 A ¢ £ ¦ § = ¦ § ¦ § ¨ © & A diagonal matrix is one where all the elements off the main diagonal are zero & An identity matrix is a diagonal matrix with its leading diagonal elements all equal to one Upper Triangular and Lower Triangular Matrices 2 1 3 8 7 1 A ¢ £ ¦ § = ¦ § ¦ § ¨ © 1 2 5 9 6 1 A ¢ £ ¦ § = ¦ § ¦ § ¨ © & An upper triangular matrix is a square matrix with all elements below the leading diagonal equal to zero & A lower triangular matrix is a square matrix with all elements above the leading diagonal equal to zero & Note that a diagonal matrix is both upper and lower triangular Zero Matrix A ¢ £ = ¦ § ¨ © & A zero matrix has all elements equal to zero MATRIX ADDITION AND SUBTRACTION 1. Check that the matrices are in the same order 2. Add/subtract the elements of A to/from the corresponding elements of B Some addition properties to remember... Commutative Law: A+B = B+A Associative Law: A+(B+C) = (A+B)+C EXAMPLE If 1 4 6 5 A & ¡ = ¤ ¥ ª « and 4 2 5 9 B & ¡ = ¤ ¥ ª « , find (a) A+B and (b) B – A (a) 1 4 4 2 1 4 4 2 5 6 6 5 5 9 6 5 5 9 11 14 A B + + & ¡ & ¡ & ¡ & ¡ + = + = = ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥ + + ª « ª « ª « ª « (b) 4 2 1 4 4 1 2 4 3 2 5 9 6 5 5 6 9 5 1 4 B A--- & ¡ & ¡ & ¡ & ¡- =- = = ¤ ¥ ¤ ¥ ¤ ¥ ¤ ¥--- ª « ª « ª « ª « TRANSPOSE OF A MATRIX (A T ) 1. Swap the position of the row elements with the column elements 2. The order of the matrix also swaps whereby { m x n } = { n x m } T Some transposing properties to remember......
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Week 09 - QMA 2006S2 (Matrices Part 1) - Jessica Chung&...

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