slide1
Download
Skip this Video
Download Presentation
化工應用數學

Loading in 2 Seconds...

play fullscreen
1 / 40

化工應用數學 - PowerPoint PPT Presentation


  • 131 Views
  • Uploaded on

化工應用數學. Matrices. 授課教師: 林佳璋. Matrix. matrix. Each symbol such as a 11 , a 2n , or a mn is called an element of the matrix. The square bracket enclosing the array signifies that it is a matrix.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' 化工應用數學' - thanos


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

化工應用數學

Matrices

授課教師: 林佳璋

slide2

Matrix

matrix

Each symbol such as a11, a2n, or amn is called an element of the matrix.

The square bracket enclosing the array signifies that it is a matrix.

The two straight lines enclosing a square array signify that to be a determinant.

The matrix A contains m rows and n columns.

A is called an m by n matrix, or A is said to be of order “m×n”.

2 by 3 matrix implying that there are two rows and three columns

slide3

Type of Matrix

row matrix-

having n element arranged in a single row

column matrix-

having n element arranged in a single column

unit column matrix

slide4

Type of Matrix

square matrix-

having the number of rows of elements equivalent to the number of columns of elements.

diagonal matrix-

all the elements except those in the diagonal from the top left-hand corner to the bottom right-hand corner are zero.

unit matrix-

a diagonal matrix in which the diagonal elements are all unity.

unit matrix of order three

null matrix-

all elements in matrix are zero, and is denoted by 0.

slide5

Matrix Algebra

Matrix Addition

1.This operation can only be carried out on matrices of the same order.

2.The sum of two matrices is obtained by adding together the corresponding

elements of each matrix.

Ex:

Note:the sum or difference does not exist if the matrices are of different order

slide6

Matrix Algebra

Scalar Multiplication

If any matrix is multiplied by a number, the elements of the matrix so formed are the products of the number and the corresponding elements

of the original matrix.

Matrix Multiplication

The product of two matrices will exist only when the number of columns of the first matrix is equal to the number of rows in the second matrix. When this condition is satisfied the two matrices are said to be

“conformable” and they yield a product.

slide7

Matrix Algebra

If the matrix A is of order (m×n) and the matrix B is of order (n×m) the product C of AB=C is a matrix of order (m×n).(n×m)=(m×m). On the other hand, the product D of BA=D is matrix of order (n×m).(m×n)=(n×n).

(i) C is obtained by pre-multiplying B by A.

(ii) C is obtained by post-multiplying A by B.

slide8

Matrix Algebra

Matrix Multiplication

In the exceptional case when the products of two square matrices are equal, i.e. AB=BA, the matrices are said to commute, or be commutable.

Note:

slide9

Determinants of Square Matrices and Matrix Products

The determinant of the product of two square matrices is equal to the product of their determinants.

Even though in general ABBA

Note:

If A is a square matrix and , the matrix is called a “singular matrix”. On the other hand, if , the matrix is “non-singular”.

slide10

Transpose of a Matrix

If the rows of an (n×m) matrix are written in the form of columns, a new matrix of order (m×n) will be formed. This new matrix is called

the “transpose” of the original matrix.

Note:

A matrix and its transpose are always conformable for multiplication and

if AA’=I, the matrix A is said to be “orthogonal”.

Note:

The transpose of the product of two matrices is equal to the product of

their transposes taken in the reverse order.

slide11

Adjoint Matrices

If aij is any element of a sqaure matrix of order (nn), and the cofactor of aij in the determinant is Aij, the transpose of the matrix whose elements

are made up of all the Aijs is called the “adjoint of A”, and is written as

slide12

Inverse of a Square Matrix

Consider a square matrix A to be non-singular and of order (nn)

Define a new matrix

slide16

Solution of Linear Algebraic Equations

x1=5.704, x2=0.261, x3=1.452, x4=1.313

slide17

Matrix Series

Powers of Matrices

Matrix Polynomials

slide18

Matrix Series

X is a square matrix

slide20

Lambda-Matrices

A -matrix is a square matrix in which the elements are function of a scalar parameter .

when =1 the determinant of the matrix () is zero, and hence this matrix is singular. However, its minors of order two are non-

vanishing, therefore the -matrix is of rank 2.

slide21

Characteristic Equation

Let A be a square matrix of order n whose elements are all constants, and let x be a column whose elements are related by the equation

where  is a scalar coefficient.

where (A-I) is a square -matrix of order n.

characteristic equation of matrix A

Roots p are called the “latent roots” or the “eigenvalues” of A.

Note: A and its transpose have the same latent roots.

slide22

Diagonal Canonical Form

The vector corresponding to the latent root is called a “latent column vector”. Let k1, k2, k3, …,kn be the latent column vectors corresponding to the latent roots of A, and K the square matrix comprising all the column

vectors. K will be of order n and non-singular.

Showing that the matrix A can be reduced to what is known as “diagonal canonical form”

slide23

Example

Evaluate the latent roots and reduce the following matrix to diagonal form

slide27

Cayley-Hamilton Theorem

Any square matrix satisfies its own characteristic equation.

The characteristic equation of the above matrix is

slide28

Cayley-Hamilton Theorem

Calculate A4 if

The characteristic equation of the matrix A is

Replacing  by A and rearranging gives

Multiplying by A gives

Multiplying by A gives

slide29

Example

Consider the matrix

Find

(a) A1/2=?

(b) sinA=?

(c) exp(A2)=?

slide31

Application of Matrix

System of first order linear ordinary differential equation

n=1

n>1

slide32

Example

Solve

slide33

Example

Solve

slide35

Application of Matrix

Consider system of first order non-linear ordinary differential equation

Let

slide36

Example

Solve

slide38

Application of Matrix

System of second order homogeneous ordinary differential equation

Let

slide39

Example

Solve

Let