Download presentation

Presentation is loading. Please wait.

Published byPreston Briggs Modified over 2 years ago

1
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix – Basic Definitions Chapter 3 Systems of Differential Equations

2
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix – Properties Matrices A, B and C with elements a ij, b ij and c ij, respectively. 1. Equality For A and B each be m by n arrays Matrix A = Matrix B if and only if a ij = b ij for all values of i and j. 2. Addition A + B = C if and only if a ij + b ij = c ij for all values of i and j. For A, B and C each be m by n arrays 3. Commutative A + B = B + A 4. Associative (A + B) + C = A + (B + C) If B = O (the null matrix), for all A : A + O = O + A = A 5. Multiplication (by a Scalar) αA = (α A)in which the elements of αA are α a ij Chapter 3 Systems of Differential Equations

3
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product if and only if Matrix multiplication * In general, matrix multiplication is not commutative ! commutator bracket symbol But if A and B are each diagonal * associative * distributive The product theorem For two n × n matrices A and B Chapter 3 Systems of Differential Equations

4
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product Successive multiplication of row i of A with column j of B – row by column multiplication For example : [2 × 3] × [3 × 2] = [2 × 2] Chapter 3 Systems of Differential Equations

5
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Multiplication, Inner Product For example : [3 × 2] × [2 × 2] = [3 × 2] Chapter 3 Systems of Differential Equations

6
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Unit Matrix, Null Matrix The unit matrix 1 has elements δ ij, Kronecker delta, and the property that 1A = A1 = A for all A The null matrix O has all elements being zero ! Exercise 3.2.6(a) : if AB = 0, at least one of the matrices must have a zero determinant. If A is an n × n matrix with determinant 0, then it has a unique inverse A -1 so that AA -1 = A -1 A = 1. Chapter 3 Systems of Differential Equations

7
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Direct product --- The direct tensor or Kronecker product If A is an m × m matrix and B an n × n matrix The direct product C is an mn × mn matrix with elements with For instance, if A and B are both 2 × 2 matrices The direct product is associative but not commutative ! Chapter 3 Systems of Differential Equations

8
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Diagonal Matrices If a 3 × 3 square matrix A is diagonalIn any square matrix the sum of the diagonal elements is called the trace. 1. The trace is a linear operation : 2. The trace of a product of two matrices A and B is independent of the order of multiplication : (even though AB BA) 3. The trace is invariant under cyclic permutation of the matrices in a product. Chapter 3 Systems of Differential Equations

9
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Inversion Matrix A An operator that linearly transforms the coordinate axes Matrix A -1 An operator that linearly restore the original coordinate axes The elements Where C ji is the jith cofactor of A. For example : The cofactor matrix C and Chapter 3 Systems of Differential Equations

10
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Matrix Inversion For example : |A| = (3)(-1-0)-(-1)(-2-0)+(1)(4-1) = -2 The elements of the cofactor matrix are Chapter 3 Systems of Differential Equations

11
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Special matrices A matrix is called symmetric if: A T = A A skew-symmetric (antisymmetric) matrix is one for which: A T = -A An orthogonal matrix is one whose transpose is also its inverse: A T = A -1 Any matrix symmetricantisymmetric Chapter 3 Systems of Differential Equations

12
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Inverse Matrix, A -1 The reverse of the rotation Transpose Matrix, Defining a new matrix such that holds only for orthogonal matrices ! Chapter 3 Systems of Differential Equations

13
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues A is a matrix, v is an eigenvector of the matrix and λ the corresponding eigenvalue. This only has none trivial solutions for det (A- λ I) = 0. This gives rise to the secular equation for the eigenvalues: Chapter 3 Systems of Differential Equations

14
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues Chapter 3 Systems of Differential Equations

15
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Eigenvectors and Eigenvalues Chapter 3 Systems of Differential Equations

16
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Example 3.5.1 Eigenvalues and Eigenvectors of a real symmetric matrix The secular equation λ = -1,0,1 λ = -1. x+y = 0, z = 0 Normalized λ = 0 x = 0, y = 0 λ = 1 -x+y = 0, z = 0 Normalized Chapter 3 Systems of Differential Equations

17
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Example 3.5.2 Degenerate Eigenvalues The secular equation λ = -1,1,1 λ = -1. 2x = 0, y+z = 0 Normalized λ = 1 -y+z = 0 (r 1 perpendicular to r 2 ) λ = 1 Normalized (r 3 must be perpendicular to r 1 and may be made perpendicular to r 2 ) Chapter 3 Systems of Differential Equations

18
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Conversion of an nth order differential equation to a system of n first-order differential equations Setting,,, …… ……

19
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Example : Mass on a spring assume eigenvector

20
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Homogeneous systems with constant coefficients in components y 1 y 2 -plane is called the phase plane Critical point : the point P at which dy 2 /dy 1 becomes undetermined is called P : (y 1,y 2 ) = (0,0)

21
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Five Types of Critical points

22
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Criteria for Types of Critical points P is the sum of the eigenvalues, q the product and the discriminant.

23
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Stability Criteria for Critical points

24
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations Example : Mass on a spring p = -c/m, q = k/m and = (c/m) 2 -4k/m No damping c = 0 : p = 0, q > 0 a center Underdamping c 2 0, < 0 a stable and attractive spiral point. Critical damping c 2 = 4mk : p 0, = 0 a stable and attractive node. Overdamping c 2 > 4mk : p 0, > 0 a stable and attractive node.

25
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations No basis of eigenvectors available. Degenerate node If matrix A has a double eigenvalue since If matrix A has a triple eigenvalue

26
Y.M. Hu, Assistant Professor, Department of Applied Physics, National University of Kaohsiung Chapter 3 Systems of Differential Equations No basis of eigenvectors available. Degenerate node

Similar presentations

Presentation is loading. Please wait....

OK

Computer Graphics Recitation 5.

Computer Graphics Recitation 5.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

What does appt only mean Ppt on new product development steps Ppt on hydrogen fuel rotary engine Ppt on kingdom monera example Ppt on tamper resistant bolts Ppt online open enrollment Ppt on formal education Ppt on product specification form Ppt on job rotation evaluation Ppt on sources of energy for class 8th paper