lecture 3 matrix and array operators

8/3/2019 Lecture 3 Matrix and Array Operators

1/51

2

Contents

Matrix and ArrayMatrix and vector

Array and matrix operations

Solving linear systems of eq.

Polynomial and its operation

Relational & logical operators

Some conversion functions

The precedence of operators


2/51

3.1 Matrix and Vector

MATLAB is a matrix-based computing environment.All of the data that you enter into MATLAB is stored

in the form of a matrix or a two-dimensional array.

Row Vector is a matrix of 1-by-n .

Column Vector is a matrix of n-by-1.


3/51

3.1.1 Creating Matrix

1. The simplest way to create a matrix in MATLAB is

to use the matrix constructor operator, []. For example,

M = [row1;row2;row3;;rown];

Row vector RV = [e1,e2,,en];

Column vector CV = [e1;e2;;en];

Empty matrix EM= [];


4/51

3.1.1 Creating Matrix

2. Using Colon operator(:) and shortcut expression:

first : incr : last .

For example

1. V = 1:10;2. V2 = 1:0.5:10;3. m = [1:4;2:5;3:6;4:7];


5/51


6/51

Table3.1 Matrix Functions

Function Description

ones Create a matrix or array of all ones.

zeros Create a matrix or array of all zeros.

eye Create a matrix with ones on the diagonal

and zeros elsewhere.

diag Create a diagonal matrix from a vector.

rand Create a matrix or array of uniformly

distributed random numbers.randn Create a matrix or array of normally

distributed random numbers and arrays.


7/51

Table3.1 Matrix Functions (continued)


magic Create a square matrix with rows, columns,and diagonals that add up to the samenumber.

randperm Create a vector (1 by n matrix) containing a

random permutation of the specifiedintegers.

Note: you can use the help elmat to list all the functions.

See : example chpt3_1.m


8/51

diag ()

X = diag(v,k) when v is a vector of n components, returns

a square matrix X of order n+abs(k), with the elements of

v on the kth diagonal. k = 0 represents the main diagonal,

k > 0 above the main diagonal, and k < 0 below the main

diagonal.

X = diag(v) puts v on the main diagonal, same as above

with k = 0.

v = diag(X,k) for matrix X, returns a column vector vformed from the elements of the kth diagonal of X.

v = diag(X) returns the main diagonal of X, same as

above with k = 0.

See Exam le ch t3-1a.m


9/51

3.1.2 Matrix concatenate

1. The brackets [] operator discussed earlier serves not

only as a matrix constructor, but also as the MATLAB

concatenation operator. The expression C = [A B]

horizontally concatenates matrices A and B.>> a = [1:3;2:4;3:5];

>> b=ones(3,2);

>> c = [a,b]

c =

1 2 3 1 1

2 3 4 1 1

3 4 5 1 1


10/51


Using Matrix Concatenation Functions

Note: the number of rows or columns of two matricsmust be the same.


cat Concatenate matrices along the specifieddimension

vertcat Vertically concatenate matrices

horzcat Horizontally concatenate matrices


11/51


Cat: Concatenate arrays along specified dimension

C = cat(dim, A, B)

concatenates the arrays A and B along dim.

C = cat(dim, A1, A2, A3, A4, ...)

concatenates all the input arrays (A1, A2, A3, A4, and so

on) along dim.

For example:

cat(2, A, B) is the same as [A, B], and cat(1, A, B) is the

same as [A; B].

See Example chpt3_1b.m


12/51

For example

>> a = [1 2 3; 2 3 4;3 4 5];

>> b = [ 1 1;1 1; 1 1];

>> d = cat(2,a,b)

d =

1 2 3 1 1

2 3 4 1 1

3 4 5 1 1

>> d2=horzcat(a,b)

d2 =

1 2 3 1 1

2 3 4 1 1

3 4 5 1 1


13/51

3.1.3 Matrix Transpose

For real matrices, the transpose operation interchanges

aij and aji. MATLAB uses the apostrophe (or single quote)

to denote transpose.

For example,

>> A = magic(3)

A =

8 1 6

3 5 7

4 9 2

>> A'ans =

8 3 4

1 5 9

6 7 2

Transposition turns a row vector into a column vector.


14/51

3.1.3 Matrix Transpose

For a complex vector or matrix, z, the quantity z'

denotes the complex conjugate transpose, where the

sign of the complex part of each element is reversed.

The unconjugated complex transpose, where the

complex part of each element retains its sign, is

denoted by z. .

See example.


15/51

Example : Transpose of Complex vector

>> Z = [ 1+2*i 2+3*i 3+4*i]

Z =1.0000 + 2.0000i 2.0000 + 3.0000i 3.0000 + 4.0000i

>> Z

ans =

1.0000 - 2.0000i

2.0000 - 3.0000i3.0000 - 4.0000i

>> Z.

ans =

1.0000 + 2.0000i

2.0000 + 3.0000i

3.0000 + 4.0000i


16/51

Table3.2 Functions of getting matrix information

Function Descriptionlength Return the length of the longest dimension.

(The length of a matrix or array with any zerodimension is zero.)

ndims Return the number of dimensions.numel Return the number of elements.

size Return the length of each dimension.


17/51

For example

>> length(d)

ans =5

>> ndims(d)

ans =

2

>> numel(d)

ans =

15

>> size(d)

ans =

3 5

Here d is,

d =

1 2 3 1 1

2 3 4 1 1

3 4 5 1 1


18/51


19/51

3.2 Array and Matrix Operations(2)

MATLAB uses a special symbol to distinguish array

operations from Matrix operations.

MATLAB uses a period before symbol to indicate an

array operator. For example, Matrix Multiplication

operator is * , and Array Multiplication operator is

.* .

The following slides show the list of operators.


20/51


21/51

(2) Array and Matrix Multiplication

operator Description

ArrayMultiplication

a.*b

.* Element-by-elementmultiplication of a and b.

Both arrays must be the sameshape,or one of them must be

a scalar.

MatrixMultiplication

a*b

* Matrix multiplication of a and b.

The number of column in amust be equal to the number

of row in b, or one of themmust be a scalar.


22/51

(3) Array Division

Note : Both arrays must be the same shape, or one of them is a scalar.


Array Right Divisiona./b

./ Element-by-element

division of a and b:a(i,j)/b(i,j)

Array Left Division

a.\b

.\ Element-by-elementdivision of a and b:

but with b in thenumerator:

b(i,j)/a(i,j).


23/51

(4) Matrix Division

Operator Description

Matrix

Right Divisiona / b

/ Matrix division defined bya*inv(b), where inv(b) is theinverse of matrix b.

Matrix

Left Divisiona \ b

\ Matrix division defined byinv(a)*b, where inv(a) is theinverse of matrix a.


24/51

(5) Matrix and Array Power


Array Power

a.^b

.^ Element-by-elementexponentiation of a and b:

a(i,j)^b(i,j).

Both arrays must be the sameshape,or one of them is ascalar.

Matrix powerM^p

^ Calculate the A

p

See chpt3_1.m


25/51

3.3 Solving Linear Systems of Equations

3.3.1 For Square Systems

Ax=b

Where A is a nn nonsingular square coefficient

matrix, x and b both are n 1 column vector.

The solution is

X=A\b

OR X = inv(A)*b

X = A^-1 *b;


26/51

Example Solving the linear system

% sample of Matrix left division operator.

% M-file name is leftdvs.m

A = [1 2 1 4;2 0 4 3;4 2 2 1;-3 1 3 2];

B = [ 13 28 20 6]';

X = A\B;

Y = X';

disp('X='); disp(X);

str = num2str(Y); disp(['The X is ',str]);


27/51

The result after running M-file

The solution is

>> leftdvs

X=

3.0000

-1.0000

4.0000

2.0000

The X is 3 -1 4 2


28/51


3.3.2. Overdetermined Systems: Overdetermined systems

of simultaneous linear equations are often encountered in

various kinds of curve fitting to experimental data.

For example,

(1) The experimental datat y

0.0 0.82

0.3 0.72

0.8 0.63

1.1 0.601.6 0.55

2.3 0.50

(2) The curve model is y(t) =c1+c2e-t


29/51


(3)create the Coefficient Matrix E

E =[ones(size(t)) exp(-t)]

(4)Solving the Over-determined Systems

E*c = y

c =E\y

(5) Plotting the fitting curve.

See M-file chpt3_2.m


30/51


31/51


32/51

3.4 Polynomial and its operation

For polynomial of degree n

P(x)=a0xn+a1x

n-1++an-1x+an

MATLAB denotes the coefficient vector as

P =[a0 a1 an-1 an ]

MATLAB provides a set of following functions to

manipulate the polynomials.


33/51

3.4.1 Polynomial functions

1) roots - Find polynomial roots.

2) poly - Convert roots to polynomial.

3) polyval - Evaluate polynomial.

4) polyvalm - Evaluate polynomial with matrix argument.

5) polyfit - Fit polynomial to data.

6) polyder - Differentiate polynomial.

7) polyint - Integrate polynomial analytically.

8) conv - Multiply polynomials.

9) deconv - Divide polynomials.


34/51

3.4.2 Using Polynomials functions

Consider P(x) = -0.02x3+0.1x2-0.2x+1.66

a) Find the value of p(4) and the value of P(x) at x = [1 2

3 4 5]

b) Find the roots of P(x) = 0

c) Plot the curve of P(x) at interval [1,6]

d) Find the definite integral of P(x) taken over [1,4]

e) Find P(4)

See chpt3_3.m


35/51

Curve fitting with polyfit()

MATLAB provides two functions formodeling your data by using a polynomial

function.

Polynomial Fit Functions

polyfit(x,y,n) finds the coefficients of a polynomial p(x)

of degree n that fits the data y by minimizing the sum

of the squares of the deviations of the data from the

model (least-squares fit).


36/51

Example of function polyfit()

Suppose we measure a quantity y at several values of

time t:

t : 0 0.3 0.8 1.1 1.6 2.3

y : 0.5 0.82 1.14 1.25 1.35 1.40

The data can be modeled by a polynomial function

y=a0t2+a1t+a2

Using least square fit method to find the polynomial

coefficients a0,a1,a2 with function

p=polyfit(t,y,2)

See chpt3_4.m ,that shows the function polyfit.


37/51

Example of function polyfit()


38/51

The result of polyfit()


39/51

3.5 Relational and logical operators

3.5.1 Relational operators

The relational expression has the form a1 op a2 .

Where a1 ,a2 are arithmetic expression,variables or string.

Theop is one of the following relational operators.

== equal to ~= Not equal to

> greater than >= greater than or equal to

< less than


40/51

3.5 Relational & Logical Operators

3.5.2 logical operators

The logical expression has the forms.

l1 op l2 or op l1

Where the l1 and l2 are logical expression or variables,and

op is the one of the logical operators shown in the follows.

~ logical NOT & logical AND

| logical OR xor logical Exclusive OR

MATLAB interprets a zero value as false, and any nonzero

values as true.


41/51

3.5 Relational & Logical Operators

3.5.3 Examples


42/51

3.5.4 Using Logicals in Array Indexing

The logical vectors created from logical and relational

operations can be used to reference subarrays.

Suppose X is an ordinary matrix and L is a matrix of

the same size that is the result of some logicaloperation. Then X(L) specifies the elements of X where

the elements of L are nonzero.


43/51

3.5.4 Examples of Using Logical indexing

Logical Indexing Example 1. This example creates

logical array B that satisfies the condition A > 0.5, and

uses the positions of ones in B to index into A. This is

called logical indexing:

>> A = rand(5); %create a 55 matrix(array).

>> B = A > 0.5; %generate a same size logical matrix(array).

>> A(B) = 0 %Logical indexing.

A =

0.2920 0.3567 0.1133 0 0.0595

0 0.4983 0 0.2009 0.0890

0.3358 0.4344 0 0.2731 0.2713

0 0 0 0 0.4091

0.0534 0 0 0 0.4740

See Example chpt3_6.m


44/51

3.5.4 Examples of Using Logical indexing

Logical Indexing Example 2. This example highlights the location of theprime numbers in a magic square using logical indexing to set the nonprimes

to 0:>> A = magic(4)

A =

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1>> B = isprime(A)

B =

0 1 1 1

1 1 0 0

0 1 0 0

0 0 0 0

>> A(~B) = 0; % Logical indexing>> A

A =

0 2 3 13

5 11 0 0

0 7 0 0

0 0 0 0


45/51

Logical Indexing Example 2

>> A = magic(4)

A =

16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

>> A(~isprime(A))=0 %Logical indexing

A =

0 2 3 13

5 11 0 0

0 7 0 0

0 0 0 0


46/51

3.6 Some Conversion Functions

1. int8, int16, int32, int64: Convert to signed integer.

I = int?(X) converts the elements of array X into signed integers. double

and single values are rounded to the nearest int? value on conversion.

2. ceil Round toward infinity.

B=ceil(A) rounds the elements of A to the nearest integers greater than

or equal to A.

3. fix Round towards zero.

B = fix(A) rounds the elements of A toward zero, resulting in an array

of integers.


47/51

3.6 Some Conversion Functions

4. floor Round towards minus infinity.

B = floor(A) rounds the elements of A to the nearest integers less

than or equal to A.

5. round Round to nearest integer.

Y = round(X) rounds the elements of X to the nearest integers.

6. mod - Modulus (signed remainder after division).

7. rem - Remainder after division.

See Example chpt3-7.m


48/51

3.7 The precedence of Operators

1. Parentheses ()

2. Transpose (.'), complex conjugate transpose ('), matrix power (^) ,

power (.^)

3. Unary plus (+), unary minus (-), logical negation (~)

4. Multiplication (.*), right division (./), left division(.\), matrix

multiplication (*), matrix right division (/), matrix left division (\)

5. Addition (+), subtraction (-)

6. Colon operator (:)

7. Less than (=), equal to (==), not equal to (~=)

8. Element-wise AND (&)

9. Element-wise OR (|)

H k 2


49/51

Homework 2

1 Set up a vector called N with five elements having the values: 1, 2, 3,

4, 5. Using N, create assignment statements for a vector X whichwill result in X having these values:

a) 2, 4, 6, 8, 10

b) 1/2, 1, 3/2, 2, 5/2

c) 1, 1/2, 1/3, 1/4, 1/5d) 1, 1/4, 1/9, 1/16, 1/25

2.The identity matrix is a square matrix that has ones on the

diagonal and zeros elsewhere. You can generate one with the

eye() function in MATLAB. Use MATLAB to find a matrix B,

such that when multiplied by matrix A=[ 1 2; -1 0 ] the identitymatrix I=[ 1 0; 0 1 ] is generated. That is A*B=I.

(tips: use "/" operator to get B)

Send your answers to [email protected] before Nov 4, 2010

H k 2
mailto:[email protected]:[email protected]


50/51

3 Using left division operator ('\') to solve curve fitting problem.

a)Find the least squares parabola f(x) = ax2+bx+c for the set of data

b)Compare your results (a,b,c) with the results returned by built-in

function p=polyfit(x,y,2).

c)Plot the sample data points with blue circle marker and the fitting

curve.

Homework 2

xk 2 1 0 1 2

yk 5.8 1.1 3.8 3.8 1.5

Send your answers to [email protected] before Nov 4, 2010
mailto:[email protected]:[email protected]


51/51

Thanks

lecture 3 matrix and array operators

Documents