CITS2401 Computer Analysis & Visualisation

Transcription

1 FACULTY OF ENGINEERING, COMPUTING AND MATHEMATICS CITS2401 Computer Analysis & Visualisation SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING Topic 7 Matrix Algebra Material from MATLAB for Engineers, Moore, Chapters 10 Additional material by Peter Kovesi and Wei Liu

2 Objectives After this lecture you should be able to: Perform the basic operations of matrix algebra Understand linear transformations Solve simultaneous equations using MATLAB matrix operations Use some of MATLAB s special matrices

3 The difference between an array and a matrix Most engineers use the two terms interchangeably The only time you need to be concerned about the difference is when you perform matrix algebra calculations Arrays: Technically an array is an orderly grouping of information Arrays can contain numeric information, but they can also contain character data, symbolic data etc. Matrices: The technical definition of a matrix is a two-dimensional numeric array used in linear algebra Not even all numeric arrays can precisely be called matrices - only those upon which you intend to perform linear transformations meet the strict definition of a matrix.

4 Matrix Operations and Functions Matrix algebra is used extensively in engineering applications Matrix algebra is different from the array calculations we have performed thus far

5 Array Operators A.* B multiplies each element in array A times the corresponding element in array B A./B divides each element in array A by the corresponding element in array B A.^B raises each element in array A to the power in the corresponding element of array B A+B adds each element in array A to the corresponding element in array B A-B subtracts each element in array A from the corresponding element in array B

6 Operators used in Matrix Mathematics Transpose Multiplication Division Exponentiation Left Division Some Matrix Algebra functions Dot products Cross products Inverse Determinants

7 Transpose In mathematics texts you will often see the transpose indicated with superscript T A T The MATLAB syntax for the transpose is A'

8 !!!! " # $ $ $ $ % & = A!!! " # $ $ $ % & = T A The transpose switches the rows and columns

9 Using the transpose with complex numbers When used with complex numbers, the transpose operator returns the complex conjugate

10 Complex Conjugate Complex numbers have a real part, which is a normal number, and an imaginary part which is a number multiplied by i, the square root of -1. Given a complex number a + bi it s complex conjugate is a bi. This number has many of the same algebraic properties as a+ bi. Given a matrix of complex numbers, the conjugate transpose generalizes many of the properties of a complex conjugate.

11 Dot Products The dot product is sometimes called the scalar product the sum of the results when you multiply two vectors together, element by element. Equivalent statements

12 Example Calculating the Center of Gravity Finding the center of gravity of a structure is important in a number of engineering applications The location of the center of gravity can be calculated by dividing the system up into small components. xw = x 1 W 1 + x 2 W 2 + x 3 W 3 + etc... yw = y 1 W 1 + y 2 W 2 + y 3 W 3 + etc... zw = z 1 W 1 + z 2 W 2 + z 3 W 3 + etc...

13 In a rectangular coordinate system x, y, and z are the coordinates of the center of gravity W is the total mass of the system x1, x2, and x3 etc are the x coordinates of each system component y1, y2, and y3 etc are the y coordinates of each system component z1, z2, and z3 etc are the z coordinates of each system component and W1, W2, and W3 etc are the weights of each system component

14 In this example We ll find the center of gravity of a small collection of the components used in a complex space vehicle Item x, meters y, meters z meters Mass Bolt gram screw gram nut gram bracket gram Formulate the problem using a dot product

15 Input and Output Input Location of each component in an x-y-z coordinate system in meters Mass of each component, in grams Output Location of the center of gravity

16 Hand Example Find the x coordinate of the center of gravity Item x, meters Mass, gram x * m, gram meters Bolt 0.1 x 3.50 = 0.35 screw 1 x 1.50 = 1.50 nut 1.5 x 0.79 = bracket 2 x 1.75 = 3.5 sum

17 We know that The x coordinate is equal to So x x = 3 x i m i i=1 m Total = =6.535/7.54 = meters 3 x i m i i=1 3 m i i=1 This is a dot product

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20 We could use a plot to evaluate our results z-axis Center of Gravity Center of Gravity This plot was enhanced using the interactive plotting tools y-axis 0 0 x-axis

21 Matrix Multiplication Similar to a dot product Matrix multiplication results in an array where each element is a dot product. In general, the results are found by taking the dot product of each row in matrix A with each column in Matrix B

22 C i, j = N k=1 A i,k B k, j

23 Because matrix multiplication is a series of dot products the number of columns in matrix A must equal the number of rows in matrix B For an m x n matrix multiplied by an n x p matrix These dimensions must match m x n n x p The resulting matrix will have these dimensions

24 Matrix multiplication as a function

25 We could use matrix multiplication to solve the problem in Example 10.1, in a single step

26 Matrix Powers Raising a matrix to a power is equivalent to multiplying it times itself the requisite number of times A 2 is the same as A*A A 3 is the same as A*A*A Raising a matrix to a power requires it to have the name number of rows and columns

27 Matrix Inverse The inverse of a matrix A is the matrix that when multiplied by A, returns the identity matrix. MATLAB offers two approaches The matrix inverse function inv(a) Raising a matrix to the -1 power A -1

28 Equivalent approaches to finding the inverse of a matrix A matrix times its inverse is the identity matrix

29 Not all matrices have an inverse Called Singular Ill-conditioned matrices Attempting to take the inverse of a singular matrix results in an error statement The condition number of a matrix measures how close it is to being singular. The function cond(a) measures how sensitive the system of linear equations represented by the matrix A is to errors.

30 Determinants Related to the matrix inverse If the determinant is equal to 0, the matrix does not have an inverse The MATLAB function to find a determinant is det(a)

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32 Cross Products sometimes called vector products the result of a cross product is a vector always at right angles (normal) to the plane defined by the two input vectors orthogonality

33 Consider two vectors A = A x i + Ay j + Az k B = B x i + By j + Bz k The cross product is equal to A B = (A y * B z A z * B y ) i + (A z * B x A x * B z ) j + (A x B y A y B x ) k

34 Cross Products are Widely Used Cross products find wide use in statics, dynamics, fluid mechanics and electrical engineering problems

35 Linear Transformations Matrix multiplication can be visualized as a continuous linear transformation of a space. Given a two dimensional space, we can multiply even point in that space by a 2x2 matrix. This maps every point to a new point. Common transformations include rotations about the origin, scaling one or more dimensions, or change of basis.

36 Examples:

37 A script to display linear transformations

38 Examples:

39 Eigenvectors Given a linear transformation, Eigenvectors are vectors that are only transformed by a scalar factor. The corresponding scalar factor for each Eigenvector is called an Eigenvalue. A non-singular 2x2 matrix will always have two Eigenvectors, although they maybe complex vectors, as is the case for rotations.

40 Solutions to Systems of Linear Equations 3x +2y z = 10 x +3y +2z = 5 x y z = 1 Three variables and three equations should have a solution. We can solve these systems by subtracting one equation from another until we manage to eliminate all but one variable. This is a time consuming process and can lead to large errors if the matrix has a high condition number.

41 Using Matrix Nomenclature " $ A = $ $ # %! ' # ' X = # ' # & " x y z $ " & $ & B = $ & $ % # % ' ' ' & and AX=B

42 We can solve this problem using the matrix inverse approach This approach is easy to understand, but its not the more efficient computationally

43 Matrix left division Matrix left division: A\B is a more efficient alternative. It also has left round off error. A related function are rref(a) for calculating the reduced row echelon form of A. Matrix left division uses Gaussian elimination, which is much more efficient, and less prone to round-off error