1 Linear Algebraic Equations and Matrices

Three Individuals Connected by Bungee Cords

Free-body Diagrams  Newton’s second law 3 Rearrange equations [K] {x} = {b}

Newton’s Second Law – Equation of Motion  Mass-spring system (similar to bungee jumpers)  Kirchhoff’s current and voltage rules 4 Resistor circuits

 Solved a single equation  Now consider more than one variable and more than one equation Linear Algebraic Equations

Linear Systems  Linear equations and constant coefficients  a ij and b i are constants 6

Mathematical Background  Write system of equations in matrix-vector form 7

Solving Systems of Linear Equations  Two ways to solve systems of linear algebraic equations [A]{x}={b}  Left-division x = A\b  Matrix inversion x = inv(A)*b  Matrix inversion only works for square, non- singular systems  Less efficient than left-division 8

向量的乘法、除法  向量的乘法 >>x=[1 2 3]; >>pi*x % 純量乘向量 ans = >>x=[1 2 3]; >>y=[4 5 6]; >>x.*y % 向量乘向量 ans =  向量的除法運算在代數系統 實未定義，但在 MATLAB 中 x./y 表示 x.\y 表示 >>x=[1 2 3]; >>y=[4 5 6]; >>x./y % 向量除向量 ans =

Matrix Notations 10 Column 4 Row 3 (second index) (first index)

矩陣的索引與下標 11

Scalars, Vectors, Matrices  MATLAB treats variables as “matrices”  Matrix (m  n) - a set of numbers arranged in rows (m) and columns (n)  Scalar: 1  1 matrix  Row Vector: 1  n matrix  Column Vector: m  1 matrix 12

Matrix Operations  Transpose ( 轉置 )  In MATLAB, transpose is A  Trace is sum of diagonal elements  In MATLAB, trace(A) 13

Matrix Transpose 14

Special Matrices  Symmetric matrices 15 a ij = a ji [A] T = [A]

Special Matrices  Diagonal matrix 16 [A][I] = [I][A] = [A]  Identity matrix ( 單位矩陣 )

Special Matrices  Banded matrix – all elements are zero, with the exception of a band centered on the main diagonal 17 Tridiagonal – three non-zero bands

Special Matrices  Lower triangular  Upper triangular 18

 Matrix identity [A] = [B] if and only if a ij = b ij for all i and j  Matrix addition and subtraction [C] = [A] + [B]  C ij = A ij + B ij [C] = [A]  [B]  C ij = A ij  B ij  Commutative [A] + [B] = [B] + [A] [A]  [B] =  [B] + [A]  Associative ( [A] + [B] ) + [C] = [A] + ( [B] + [C] ) ( [A] + [B] )  [C] = [A] + ( [B]  [C] ) ( [A]  [B] ) + [C] = [A] + (  [B] + [C] ) Matrix Operations

g = 5 Multiplication of Matrix by a Scalar

Matrix Multiplication  Visual depiction of how the rows and columns line up in matrix multiplication 21

[A]*[B]  [B]*[A] Matrix Multiplication

 Matrix multiplication can be performed only if the inner dimensions are equal 23

Matrix Multiplication  Interior dimensions have to be equal  For a vector  We will be using square matrices 24

Matrix Multiplications  MATLAB performs matrix multiplication automatically A*B = C  Note: no period ‘.’ (not element-by-element operation)  For vectors A*x = b  Associative ( [A] [B] ) [C] = [A] ( [B] [C] )  Distributive [A] ( [B] + [C] ) = [A] [B] + [A] [C] ([A] + [B] ) [C] = [A] [C] + [B] [C]  Not generally commutative [A] [B]  [B] [A] 25

Matrix Inverse  Matrix division is undefined  However, there is a matrix inverse for non-singular square matrices [A] -1 [A] = [A] [A] -1 = [I]  Multiplication of a matrix by the inverse is analogous to division 26

Augmentation  Whatever you do to left-hand-side, do to the right- hand side (useful when solving system of equations) 27

Augmented Matrix

>> A = [ ; ; ; ] A = >> A' ans = Create a matrix Matrix transpose MATLAB Matrix Manipulations

 Matrix concatenation >> x = [ ]; >> y = [ ]; >> z = [ ]; >> B = [x; y; x; z] B = >> C = A + B C = >> C = C – B ( = A) C =

MATLAB Matrix Manipulations  Matrix multiplication, element-by-element operation 31 A = B = >> A*B ans = >> A.*B ans = A*B  A.*B

>> D = [ ; ; ] D = >> A*D ??? Error using ==> * Inner matrix dimensions must agree. >> D*A ans = A = Inner dimension must agree A*D  D*A MATLAB Matrix Manipulations

>> A = [ ; ; ; ] >> format short; AI = inv(A) AI = >> A*AI ans = Matrix Inverse MATLAB Matrix Manipulations

>> A = [ ; ; ; ] >> I = eye(4) I = >> Aug = [A I] Aug = >> [n, m] = size(Aug) n = 4 m = 8 Matrix Augmentation MATLAB Matrix Manipulations

[K] {x} = {b} JumperMass (kg)Spring constant (N/m)Unstretched cord length (m) Top (1) Middle (2) Bottom (3) Bungee Jumpers

>> k1 = 50; k2 = 100; k3 = 50; >> K=[k1+k2 -k2 0;-k2 k2+k3 -k3;0 -k3 k3] K = >> format short >> g = 9.81; mg = [60; 70; 80]*g mg = >> x=K\mg x = >> xi = [20; 40; 60]; >> xf = xi + x xf = >> x = inv(K)*mg x = k 1 = 50 k 2 = 100 stiffer cord k 3 = 50 Final positions of bungee jumpers

Kirchhoff’s Current and Voltage Rules  Example i 43 i 32 i 54 i 52 i 65 i 12

Kirchhoff’s Current and Voltage Rules  Example

>> A = [ ]; >> b = [ ]'; >> current = A\b current =