# Week 4 Linear algebraic equations 2020.pdf - Linear...

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Linear algebraic equations Dr. Elena Atroshchenko School of Civil and Environmental Engineering
Mathematical Background 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 ….. 𝑎𝑎 1𝑚𝑚 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 ….. 𝑎𝑎 2𝑚𝑚 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 ….. 𝑎𝑎 3𝑚𝑚 . . … … … . . . . … … 𝑎𝑎 𝑛𝑛1 𝑎𝑎 𝑛𝑛2 𝑎𝑎 𝑛𝑛3 ….. 𝑎𝑎 𝑛𝑛𝑚𝑚 A n by m matrix: n rows m columns
Mathematical Background 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 ….. 𝑎𝑎 1𝑚𝑚 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 ….. 𝑎𝑎 2𝑚𝑚 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 ….. 𝑎𝑎 3𝑚𝑚 . . … … … . . . . … … 𝑎𝑎 𝑛𝑛1 𝑎𝑎 𝑛𝑛2 𝑎𝑎 𝑛𝑛3 ….. 𝑎𝑎 𝑛𝑛𝑚𝑚 A n by m matrix: row 3, columns 2
Mathematical Background 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 𝐴𝐴 = 1 3 5 3 2 4 5 4 6 Symmetric matrix 𝑎𝑎 𝑖𝑖𝑖𝑖 = 𝑎𝑎 𝑖𝑖𝑖𝑖 Diagonal matrix 𝐴𝐴 = 𝑎𝑎 11 0 0 0 𝑎𝑎 22 0 0 0 𝑎𝑎 33 If n = m, matrix is called square matrix .
Mathematical Background Identity matrix 𝐴𝐴 = 1 0 0 0 1 0 0 0 1 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 0 𝑎𝑎 22 𝑎𝑎 23 0 0 𝑎𝑎 33 Upper triangular matrix Lower triangular matrix 𝐴𝐴 = 𝑎𝑎 11 0 0 𝑎𝑎 21 𝑎𝑎 22 0 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33
Mathematical Background 𝐵𝐵 = [ 𝑏𝑏 1 𝑏𝑏 2 𝑏𝑏 3 𝑏𝑏 𝑚𝑚 ] (row) vector = 1 by m matrix (column) vector = n by 1 matrix 𝐶𝐶 = 𝑐𝑐 1 𝑐𝑐 2 𝑐𝑐 3 . . . 𝑐𝑐 𝑛𝑛
Mathematical Background Addition of two matrices 𝐴𝐴 + 𝐵𝐵 = 𝑎𝑎 𝑖𝑖𝑖𝑖 + 𝑏𝑏 𝑖𝑖𝑖𝑖 Subtraction of two matrices 𝐴𝐴 − 𝐵𝐵 = 𝑎𝑎 𝑖𝑖𝑖𝑖 − 𝑏𝑏 𝑖𝑖𝑖𝑖 Commutative 𝐴𝐴 + 𝐵𝐵 = 𝐵𝐵 + [ 𝐴𝐴 ] Associative ( 𝐴𝐴 + 𝐵𝐵 ) + 𝐶𝐶 = [ 𝐴𝐴 ] + ( 𝐵𝐵 + 𝐶𝐶 ) 𝑔𝑔 𝐴𝐴 = 𝑔𝑔𝑎𝑎 11 𝑔𝑔𝑎𝑎 12 𝑔𝑔𝑎𝑎 13 𝑔𝑔𝑎𝑎 21 𝑔𝑔𝑎𝑎 22 𝑔𝑔𝑎𝑎 23 𝑔𝑔𝑎𝑎 31 𝑔𝑔𝑎𝑎 32 𝑔𝑔𝑎𝑎 33 Multiplication by scalar g
Mathematical Background Matrix multiplication 𝐶𝐶 = 𝐴𝐴 𝐵𝐵 𝑐𝑐 𝑖𝑖𝑖𝑖 = 𝑘𝑘=1 𝑛𝑛 𝑎𝑎 𝑖𝑖𝑘𝑘 𝑏𝑏 𝑘𝑘𝑖𝑖 Associative 𝐴𝐴 𝐵𝐵 [ 𝐶𝐶 ] = 𝐴𝐴 ( 𝐵𝐵 𝐶𝐶 ) Distributive 𝐴𝐴 𝐵𝐵 + 𝐶𝐶 = 𝐴𝐴 𝐵𝐵 + 𝐴𝐴 [ 𝐶𝐶 ] Not generally commutative 𝐴𝐴 𝐵𝐵 [ 𝐵𝐵 ][ 𝐴𝐴 ]
Mathematical Background Analogous to division 𝐴𝐴 𝐴𝐴 −1 = [ 𝐴𝐴 ] −1 𝐴𝐴 = [ 𝐼𝐼 ] Matrix inversion [ 𝐴𝐴 ] −1 = 1 𝑎𝑎 11 𝑎𝑎 22 − 𝑎𝑎 12 𝑎𝑎 21 𝑎𝑎 22 −𝑎𝑎 12 −𝑎𝑎 21 𝑎𝑎 11 𝐴𝐴 𝑇𝑇 = 𝑎𝑎 11 𝑎𝑎 21 𝑎𝑎 31 𝑎𝑎 12 𝑎𝑎 22 𝑎𝑎 32 𝑎𝑎 13 𝑎𝑎 23 𝑎𝑎 33 Matrix transpose 𝐴𝐴 𝑇𝑇 = 𝑎𝑎 11 𝑎𝑎 21 𝑎𝑎 31 𝑎𝑎 12 𝑎𝑎 22 𝑎𝑎 32 𝑎𝑎 13 𝑎𝑎 23 𝑎𝑎 33 1 0 0 0 1 0 0 0 1 Augmentation (addition of columns)
Determinant The 3 by 3 matrix 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 The determinant 𝐷𝐷 of this matrix is 𝐷𝐷 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 This is a matrix This is a single number
Determinant For example, a second-order matrix determinant is 𝐷𝐷 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 21 𝑎𝑎 22 = 𝑎𝑎 11 𝑎𝑎 22 − 𝑎𝑎 12 𝑎𝑎 21 For a third-order determinant 𝐷𝐷 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 = 𝑎𝑎 11 𝑎𝑎 22 𝑎𝑎 23 𝑎𝑎 32 𝑎𝑎 33 − 𝑎𝑎 12 𝑎𝑎 21 𝑎𝑎 23 𝑎𝑎 31 𝑎𝑎 33 + 𝑎𝑎 13 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 31 𝑎𝑎 32
Mathematical Background Matrix vector multiplication 𝐴𝐴 𝑋𝑋 = { 𝑌𝑌 } n by n n by 1 n by 1
Mathematical Background Matrix vector multiplication 𝐴𝐴 𝑋𝑋 = { 𝑌𝑌 } 𝑦𝑦 𝑖𝑖 = 𝑘𝑘=1 𝑛𝑛 𝑎𝑎 𝑖𝑖𝑘𝑘 𝑥𝑥 𝑘𝑘
Simultaneous linear equations How can we numerically solve simultaneous equations? 𝑎𝑎 11 𝑥𝑥 1 + 𝑎𝑎 12 𝑥𝑥 2 + + 𝑎𝑎 1𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 1 𝑎𝑎 21 𝑥𝑥 1 + 𝑎𝑎 22 𝑥𝑥 2 + + 𝑎𝑎 2𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 2 𝑎𝑎 𝑛𝑛1 𝑥𝑥 1 + 𝑎𝑎 𝑛𝑛2 𝑥𝑥 2 + + 𝑎𝑎 𝑛𝑛𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 𝑛𝑛
Simultaneous linear equations How can we numerically solve simultaneous equations? 𝑎𝑎 11 𝑥𝑥 1 + 𝑎𝑎 12 𝑥𝑥 2 + + 𝑎𝑎 1𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 1 𝑎𝑎 21 𝑥𝑥 1 + 𝑎𝑎 22 𝑥𝑥 2 + + 𝑎𝑎 2𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 2 𝑎𝑎 𝑛𝑛1 𝑥𝑥 1 + 𝑎𝑎 𝑛𝑛2 𝑥𝑥 2 + + 𝑎𝑎 𝑛𝑛𝑛𝑛 𝑥𝑥 𝑛𝑛 = 𝑏𝑏 𝑛𝑛 𝐴𝐴 𝑋𝑋 = { 𝐵𝐵 }
Simultaneous linear equations System of linear equations can be written in the matrix form: 𝐴𝐴 𝑋𝑋 = { 𝐵𝐵 } 𝐴𝐴 = 𝑎𝑎 11 𝑎𝑎 12 𝑎𝑎 13 ….. 𝑎𝑎 1𝑛𝑛 𝑎𝑎 21 𝑎𝑎 22 𝑎𝑎 23 ….. 𝑎𝑎 2𝑛𝑛 𝑎𝑎 31 𝑎𝑎 32 𝑎𝑎 33 ….. 𝑎𝑎 3𝑛𝑛 . . … … … . . . . … … 𝑎𝑎 𝑛𝑛1 𝑎𝑎 𝑛𝑛2 𝑎𝑎 𝑛𝑛3 ….. 𝑎𝑎 𝑛𝑛𝑛𝑛 𝑋𝑋 = 𝑥𝑥 1 𝑥𝑥 2 𝑥𝑥 3 .