2 Matrices - MATRICES MATRIX DEFINITION A matrix is defined as a rectangular array of quantities arranged in rows and columns a a a a 12 13 1n 11 a

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Unformatted text preview: MATRICES MATRIX DEFINITION A matrix is defined as a rectangular array of quantities arranged in rows and columns. a a .. a a 12 13 1n 11 a 21 a 22 a 23 .. a 2n a a a .. a [ A] = 31 32 33 3n .. .. .. .. .. a a .. a a m2 m3 mn m1 Aij is the quantity A of the ith row and jth column MATRIX TYPES Column Matrix All elements are in a single column 2 - 5 3 Row Matrix All elements are in a single row [3 7 - 2] Square Matrix Any matrix with equal number of columns and rows 2 1 4 7 1 0 0 2 0 MATRIX TYPES Symmetric Matrix is a square matrix where the element aij = aji 3 [A ]= 6 - 4 6 2 1 4 1 5 Diagonal Matrix is a square matrix where all the elements are zero except those along the main diagonal 3 [B ]= 0 0 0 2 0 0 0 6 Unit or Identity Matrix is a diagonal matrix where the values of all diagonal elements are 1 1 0 0 [C]= 0 1 0 0 0 1 MATRIX TYPES Lower Triangular Matrix 7 [A ]= 8 4 2 [B ]= 0 0 0 2 6 0 0 1 3 5 2 Upper Triangular Matrix 1 4 0 Null Matrix is a matrix whose elements are zero 0 [C ]= 0 0 0 0 0 0 0 0 MATRIX OPERATIONS Two matrices ( [A] and [B]) are equal if and only if the quantities in all rows and columns are equal [A] = 2 7 3 5 - 1 4 [B] = 2 7 3 5 - 1 4 MATRIX OPERATIONS - Addition Two matrices ( [A] and [B]) can be added or subtracted if the numbers of rows and columns are identical. Add or subtract the Aij quantity to or from the corresponding Bij quantity. 5 2 [ A ] = 7 - 1 - 3 4 4 1 [ B] = 6 - 1 3 2 6 [ A] + [ B] = 13 0 6 - 2 6 MATRIX OPERATIONS TRANSPOSE OF A MATRIX The transpose of the matrix [A] denoted [A]T can be had by changing the columns in [A[ to rows in [A]T as shown below. 2 5 3 [A] = 7 - 1 4 - 3 4 5 2 7 - 3 [A] T = 5 -1 4 3 4 5 And 47 34 [B] = 27 12 28 8 47 [B]T = 12 34 28 27 8 MATRIX OPERATIONS TRANSPOSED MATRICES Transposed matrices hold the following properties: {[A] + [B]}T = [A]T + [B]T {6*[A]}T = 6*[A]T {[A]*[B]}T = [B]T * [A]T MATRIX OPERATIONS Multiplication A matrix can be multiplied by a scalar by multiplying each element in the matrix by the scalar. 6 3 [A] = 1 36 18 6 * [A] = 6 4 -3 0 0 7 3 24 0 -18 42 0 18 MATRIX OPERATIONS Multiplication The results of multiplying [A] by [B] is not the same as multiplying [B] by [A]; the product of two matrices is not commutative [A]*[B] = [B]*[A] In matrix multiplication, the distributive and associative laws are valid. [A]*{[B] + [C]} = [A]*[B] + [A]*[C] [A]*{[B]*[C]} = {[A]*[B]}*[C] MATRIX OPERATIONS Multiplication Two matrices [A] and [B] can be multiplied if and only if the Any element Cij in [ C] is equal to Aik Bkj k =1 number of columns in [A] is equal to the number of rows in [B], the two matrices are conformable The resulting matrix [C] has the same number of row as [A] and the same number of columns as [B]. That is [A] * [B] = [C] (i by m) * (m by j) = (i by j) m MATRIX OPERATIONS Multiplication If [C] is the result of multiplying [A] by [B], then [C] would have the same number of row as [A] and the same number of columns as [B]. That is [A] * [B] = [C] (i by m) * (m by j) = (i by j) Any element Cij in [C] is equal to ( A )( B ) m k =1 ik kj MATRIX OPERATIONS Example of multiplying two 3 by 3 matrices 3 [A]= 2 1 2 4 3 1 5 ; 1 - 1 [B ]= 0 4 3 1 0 2 1 6 ( 3 )( 3 ) + ( 2 )( 1) + (1)( 0 ) ( 2 )( 3 ) + ( 4 )( 1) + ( 5 )( 0 ) (1)( 3 ) + ( 3 )( 1) + ( - 1)( 0 ) ( 3 )( 2 ) + ( 2 )( 1) + (1)( 6 ) ( 2 )( 2 ) + ( 4 )( 1) + ( 5 )( 6 ) (1)( 2 ) + ( 3 )( 1) + ( - 1)( 6 ) [A]* [B ]= 7 [C ]= 22 -3 ( 3 )( 1) + ( 2 )( 0 ) + (1)( 4 ) [C ]= ( 2 )(1) + ( 4 )( 0 ) + ( 5 )( 4 ) (1)( 1) + ( 3 )( 0 ) + ( - 1)( 4 ) 11 14 10 38 - 1 6 MATRIX OPERATIONS Multiplication Example of multiplying (3 by 3) by (3 by 2) matrices to yields (3 by 2) matrix [C] 5 3 1 2 4 Multiply [ A ] = 7 - 1 4 by [ B ] = 6 - 1; [ A ] [ B ] = [ C] - 3 4 5 3 5 (2)(1) + (5)(-1) + (3)(5) (2)(4) + (5)(6) + (3)(3) [ C] = (7)(4) + (-1)(6) + (4)(3) (7)(1) + (-1)(-1) + (4)(5) (-3)(4) + (4)(6) + (5)(3) (-3)(1) + (4)(-1) + (5)(5) 47 12 [ C] = 34 28 27 8 MATRIX OPERATIONS PARTITIONING A matrix can be partitioned into submatrices as follows: a11 [ A ] = a21 a31 a12 a22 a32 a13 a23 a33 [ A11 ] = [ a11 ] ; a21 [ A 21 ] = ; a31 [ A12 ] = [ a12 a14 = A11 a24 A 21 a34 a13 A12 A 22 a14 ] a24 a34 a22 [ A 22 ] = a32 a23 a33 MATRIX OPERATIONS PARTITIONING Matrix algebra applies to partitioned matrices if they are conformable. 6 5 4 [A]= 2 - 2 3 1 0 3 [A ] [A] [B]= 11 [A21 ] - 3 2 [B]= 5 6 7 1 [A12 ] B11 [A11] [B11] + [A12 ] [B21] = [ ] [ ] + [ ] [ ] [A22 ] B21 A21 B11 A22 B21 MATRIX OPERATIONS PARTITIONING (continued) -3 37 = - 6 6 28 21 [7 1 ] = 3 21 -3 [A 21 ] [B 11 ] = [1 0 ] 2 -3] = [2 6 5 [A 22 ] [B 21 ] = [3 ] [7 1 ] = [21 3 ] 6 2 [A12 ] [B 21 ] = 4 3 [A11 ] [B 11 ] = 5 -2 2 5 12 - 18 37 [A ] [B ] = - 6 2 12 - 18 -3 28 + 21 21 4 3 3 65 = 15 23 16 - 15 0 MATRIX OPERATIONS DETERMINANTS A determinant is a square array of numbers enclosed within Vertical bars as in the nth order determinant below. A = a11 a21 .. an1 a12 a22 .. an 2 .. a1n .. a2n .. .. .. ann MATRIX OPERATIONS DETERMINANTS Evaluation of a determinant leads to a single value, which can be obtained using Laplace's expansion. The procedure uses the determinant minors and cofactors). MATRIX OPERATIONS Determinant of a square matrix (2 by 2) a11 [A]= a 21 a12 a11 ; A = a 22 a 21 a12 a 22 ; A = a11a22 - a12 a 21 For example 2 [A]= 1 - 3 ; 4 A = [( 2 )( 4 ) - ( - 3 )(1) ]= 11 MATRIX OPERATIONS Determinant of a square matrix (3 by 3) S11 S12 S13 Let [S]= Sij = S21 S22 S23 S31 S32 S33 then S11 S = S 21 S31 S12 S 22 S32 S13 S 23 S33 A determinant of order n can be evaluated from the cofactors (Coij) of element Sij from minor Mij of any row I or column j as follows: S = Sij Coij j =1 n or S = Sij Coij i =1 n MATRIX OPERATIONS Determinant of a square matrix (3 by 3) 3 [ A ]= 1 9 Co 11 = Co 12 = Co 13 = 0 -1 2 1+ 1 ` 2 7; 6 -1 2 2` A = n j= 1 a ij Co ij ( - 1) 7 = ( - 1 ) 2 {(- 1 )( 6 ) - ( 7 )( 2 )}= - 20 6 7 = ( - 1 ) 3 {( )( 6 ) - ( 7 )( 9 )}= - 57 1 6 -1 2 = ( - 1 ) 4 {( )( 2 ) - ( - 1 )( 9 )}= 11 1 ( - 1 )1+ ( - 1) 1 9 1 9 1+ 3 ` A = ( 3 )( - 20 ) + ( 0 )( - 57 ) + ( 2 )( 11 ) = - 38 MATRIX OPERATIONS Determinant of a square matrix (3 by 3), alternative method ( a11 ) A = ( a12 ) ( a ) 13 ( a22 ) ( a33 ) + ( a31 ) ( a22 ) ( a13 ) + ( a23 ) ( a31 ) + - ( a32 ) ( a23 ) ( a11 ) + ( a21 ) ( a32 ) ( a33 ) ( a21 ) ( a12 ) MATRIX OPERATIONS Determinant of a (4X4) matrix [ A = 3 3 4 0 1 1 2 0 4 0 1 2 0 3 - 1 1 1 2 0 0 1 - 1 3 - 1 1 - 2 3 4 - 2 3 4 - 2 2 3 4 - 0 4 0 1 4 3 - 1 1 1 2 0 0 1 3 - 1 1 - 2 3 4 + A = ( - 1) - 2 Where the determinants of the 3 by 3 matrices can be evaluated as before. MATRIX OPERATIONS Determinant The determinant of a matrix can also be found by converting the matrix into an uppertriangular matrix using elementary row transformation and then multiplying the elements in the diagonal as shown in the example. MATRIX OPERATIONS Determinant 1 2 3 1 1 0 0 0 4 2 0 2 4 - 6 0 0 - 2 0 -1 2 - 2 4 -3 8 3 3 4 2 -3 = 1 0 0 0 3 - 2 -3 16 - 3 1 = 0 0 0 4 - 6 - 12 - 2 4 - 6 0 0 - 2 4 5 4 - 2 4 -3 0 3 -2 -7 -6 3 -2 -3 -8 = = ( 1 )( - 6 )( - 3 )( - 8 ) = 144 MATRIX OPERATIONS Matrix Inversion Inversion of a matrix can be accomplished using the Gauss-Jordan technique. First augment the given matrix with the identity matrix of the same order, then reduce the original matrix into an identity matrix. The augmented matrix is the inversion of the original matrix. MATRIX OPERATIONS Matrix inversion example 1 -1 2 [A ]= 3 0 1 ; 1 0 2 1 -1 2 1 0 3 0 1 0 1 1 0 2 0 0 Augment matrix A with the identity matrix 0 Multiply row 1 by 3 and 0 subtract it from row 2 and subtract row 1 from 1 row 3 MATRIX OPERATIONS Matrix inversion example (continued) 1 - 2 1 [ ]= 3 0 1 A 1 0 2 Augment the matrix with the identity matrix 1 - 2 1 0 0 1 Multiply row 1 by 3 and subtract it from row 2 and 3 0 1 0 1 0 subtract row 1 from row 3 1 0 2 0 0 1 MATRIX OPERATIONS Matrix inversion example (continued) - 2 1 0 0 1 1 Interchange rows 3 and 2 - - 0 3 5 3 1 0 1 0 - 0 1 0 1 1 0 0 - 2 1 0 0 1 - 0 1 Multiply row 1 by 3 and add 1 0 1 it to row 3 3 - -3 1 0 5 MATRIX OPERATIONS Matrix inversion example (continued) 1 0 0 1 0 0 - 1 1 0 1 0 0 - 0 1 1 0 1 - 3 - 2 1 0 0 1 1 0 - 0 1 1 1 3 - - 0 1 0 5 5 2 0 - 5 Divide row 3 by 5 Add row 2 to row 1 and multiply row 3 by 2 and subtract it from row 1 MATRIX OPERATIONS Matrix inversion example (continued) 1 0 0 0 1 0 0 0 1 0 -1 0 0 -1 0 2 - 5 0 1 - 5 1 - The 3 by 3 matrix on the 5 1 right hand side is the 3 inverse [A]-1 of matrix - 5 [A] 2 1 - - 5 5 0 1 = [A]-1 1 3 - - 5 5 MATRIX OPERATIONS Matrix inversion using cofactors [S ] -1 = [ Co ]T S where [Co] is a matrix consisting of cofactors Coij of the elements of [S]. For a 2nd order matrix S11 [S ]= S21 S11 [S]-1 = S21 S12 S 22 -1 S 22 [ Co ] = - S12 S22 - S21 - S 21 S11 ; Hence -S12 S11 S12 1 = S22 ( S11 S22 -S12 S21 ) MATRIX OPERATIONS Third order matrix inversion using cofactors 3 [S ] = 7 9 11 7 7 = - 7 7 11 7 11 7 7 9 9 9 9 7 9 7 9 - 7 9 3 9 - 3 7 7 9 9 9 9 7 - 7 9 3 9 3 7 11 7 7 7 7 11 [ Co ] = 50 0 - 50 0 - 54 42 - 50 42 - 16 [S ]-1 T = 1 [Co ] S 50 1 = 0 ( -300 ) - 50 0 - 54 42 - 50 42 - 16 MATRIX OPERATIONS Matrix inversion All nonsingular square matrices are invertible. If the product of two square matrices is the identity matrix, the matrices are inverses. Singular matrices do not have an inverse. If Ax = b, then A-1Ax = A-1b and x = A-1b MATRIX APPLICATION Simultaneous equations Consider the following simultaneous equations 4x1 - 2x2 + x3 = 15 -3x1 - x2 +4x3 = 8 x1- x2 + 3x3 = 13 The matrix notation is as follows: 4 -2 1 x1 4 -2 1 15 15 - - = - - 3 1 4 x2 8 or 3 1 4 8 1 -1 3 x3 1 -1 3 13 13 MATRIX APPLICATION Simultaneous equations Multiplying row 1 by 3 and adding it to 4 times row 2 yields 4 0 0 15 -10 19 77 -2 11 37 1 -2 Multiplying row 3 by 10 and subtracting it from 2 times row 2 yields 4 0 0 -2 -10 0 1 15 19 77 -72 -216 MATRIX APPLICATION Simultaneous equations Subtracting row 2 from 5 times row 1 yields 0 -14 -2 20 0 -10 19 77 0 0 -72 -216 Multiplying row 3 by (14/72) and subtracting it from row 1and multiplying row 3 by (19/72) and adding it to row 2 yields 20 0 - 10 0 0 0 0 40 0 20 -72 -216 MATRIX APPLICATION Simultaneous equations The last matrix represents the following equations: 20x1 = 40 - 10x2 = 20 -72x3 = -216 Solving the equations yields: X1 = 2 X2 = -2 X3 = 3 ...
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This note was uploaded on 02/03/2012 for the course CE 221 taught by Professor Buch during the Summer '08 term at Michigan State University.

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