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NM2_matrixopps - Matrix Definitions and Basic Matrix...

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1 Matrix Definitions and Basic Matrix Operations Reference Ayyub & McCuen Chapter 2
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2 Matrices Definitions
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3 Matrix A rectangular array of numbers Notation: Capital letter Enclosed by square brackets, [ A ] Defined by the number of rows and number of columns [ ] 3 x 3 A
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4 Elements in a matrix Each element in a matrix can be identified by its position, row then column. Notation: lower case subscript represents row,column [ ] j , i 2 , 2 1 , 2 2 , 1 1 , 1 a element a a a a A 1 row 2 row 2 Column 1 Column ú û ù ê ë é =
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5 Elements in a matrix Elements who’s indices are equal are diagonal elements, a 1,1 The diagonal goes from the top left to the lower right Other elements are off diagonal terms
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6 Matrix use, Data sets Suppose that 5 sets of data were acquired on a refrigeration test. Temperature is measured at three locations at 5 different times. 5 4 3 2 1 3 2 1 t t t t t T T T 80 25 42 88 26 45 90 27 47 97 26 55 95 25 50
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7 Matrix use, Equations Matrices are also useful in representing equation sets. EI 6 PL d L 4 EI 24 wL d L 3 4 EI 24 wL d L 3 4 2 2 1 3 2 1 3 2 1 = - θ + θ - = - θ + θ = - θ + θ [ ] { } {} b x A EI 6 PL EI 24 wL EI 24 wL d L 4 1 1 L 3 4 1 L 3 1 4 2 3 3 2 1 = ï ï ï þ ï ï ï ý ü ï ï ï î ï ï ï í ì - = ï þ ï ý ü ï î ï í ì q q ú ú ú ú ú ú û ù ê ê ê ê ê ê ë é - - -
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8 Types of Matrices Square equal number of rows and columns Diagonal non-zero entries on the diagonal, zero in all other locations Upper Triangular non-zero on diagonal and above Lower Triangular non-zero on diagonal and below
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9 Types of Matrices Identity matrix Special diagonal matrix with 1’s in all diagonal locations Null matrix matrix of all zero’s Symmetric matrix – a i,j = a j,i
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