AN ENGINEER’S GUIDE TO MATLAB 3rd Edition CHAPTER 2 VECTORS and MATRICES

Copyright © Edward B. Magrab 2009 1Copyright © Edward B. Magrab 2009

Chapter 2An Engineer’s Guide to MATLAB

AN ENGINEER’S GUIDE TO MATLAB

3rd Edition

CHAPTER 2

VECTORS and MATRICES



Chapter 2 – Objective

Introduce MATLAB syntax in the context of vectors and matrices and their manipulation.



Topics

• Definitions of Matrices and Vectors

• Creation of Vectors

• Creation of Matrices

• Dot Operations

• Mathematical Operations with Matrices Addition and Subtraction Multiplication Determinants Matrix Inverse Solution of a System of Equations



Definitions of Matrices and Vectors

An array A of m rows and n columns is called a matrix of order (mn) and is written as

11 12 1

21 22

... ...

n

m1 mn

a a ... a

a aA = (m×n)

a a

The elements of the matrix are denoted aij, where i indicates the row number and j the column number.

Square Matrix: n = m



Diagonal Matrix: When n = m and aij = 0, i ≠j; thus

11

22 ( )

0 ... 0

0

... ...

0 nn

a

aA = n n

a

Identity Matrix: Diagonal matrix with aii = 1; thus

1 0 ... 0

0 1=

... ...

0 1

I



Column Matrix - Vector:

Row Matrix - Vector:

When aij = ai1 (there is only one column), then

( )

11 1

21 2

1

1

m m

a a

a aa = = m

... ...

a a

When aij = a1j, (there is only one row), then

( )11 12 1 1 2 1 n na a a a a a a n

Default definition of a vector in MATLAB



Transpose of a Matrix and a Vector

The transpose of a matrix is denoted by an apostrophe (). If A is defined as before and W = A, then

( )

11 11 12 21 1 1

21 12 22 22

1 1

m m

n n nm mn

w = a w = a ... w = a

w = a w = aW = A = n m

... ...

w = a w = a

11 12 1

21 22

... ...

n

m1 mn

a a ... a

a aA = (m×n)

a a

Recall that



For a row vector -

( ) ( )

1

21 2If ... 1 then 1

m

m

a

aa a a a m a m

a

1

21 2( ) ( )m

m

If 1 then ... 1...

a

aa m a a a a m

a

For a column vector -



If a, x, b, … are either variable names, numbers, expressions, or strings, then vectors are expressed as either

f = [a x b …] (Blanks required)

or

f = [a, x, b, …] (Blanks optional)

Creation of Vectors

Caution about blanks: If a = h + ds, then f is written as either

f = [h+d^s x b ...]

or

f = [h+d^s, x, b, ...]

No blanks permitted within expression



Ways to Assign Numerical Values to the Elements of a Vector -

• Specify the range of the values and the increment between adjacent values – called colon notation.

Used when the increment is either important or has been specified.

• Specify the range of the values and the number of values desired.

Used when the number of values is important.



If s, d, and f are any combination of numerical values, variable names, and expressions, then the colon notation that creates a vector x is either

x = s:d:f or x = (s:d:f) or x = [s:d:f]

where

s = start or initial valued = increment or decrementf = end or final value

Thus, the following row vector x is created

x = [s, s+d, s+2d, …, s+nd] (Note: s+nd f )



Also, when d is omitted MATLAB assumes that d = 1. Then

x = s:f

creates the vector

x = [s, s+1, s+2, … , s+n]

where s+n f

The number of terms, called the length of the vector, that this expression has created is determined from

length(x)



Example –

Create a row vector that goes from 0.2 to 1.0 in increments of 0.1.

x = 0.2:0.1:1n = length(x)

When executed, we obtainx = 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00n = 9

To create a column vector, we use

x = (0.2:0.1:1)n = length(x)

x = 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 1.0000n = 9



Generation of n Equally Spaced Values

x = linspace(s, f, n)

where the increment (decrement) is computed by MATLAB from

The values of s and f can be either positive or negative and either s > f or s < f. When n is not specified, it is assigned a value of 100. Thus, linspace creates the vector

x = [s, s+d, s+2d, …, f = s+(n1)d]

1

f sd =

n



If equal spacing on a logarithmic scale is desired, then

x = logspace(s, f, n)

where the initial value is xstart = 10s, the final value is xfinal = 10f and d is defined above.

This expression creates the row vector

x = [10s 10s+d 10s+2d … 10f] 1

f sd =

n



Examples -

x = linspace(0, 2, 5)

gives

x =

0 0.5000 1.0000 1.5000 2.0000

and

x = logspace(0, 2, 5) % [100, 100.5, 101, 101.5, 102]

gives

x =

1.0000 3.1623 10.0000 31.6228 100.0000



Transpose of a Vector with Complex Values

If a vector has elements that are complex numbers, then taking its transpose also converts the complex quantities to their complex conjugate values.

Consider the following script

z = [1, 7+4j, -15.6, 3.5-0.12j];w = z

which, upon execution, gives

w = 1.0000 7.0000 - 4.0000i -15.6000 3.5000 + 0.1200i



If the transpose is desired, but not the conjugate, then the script is written as

z = [1, 7+4j, -15.6, 3.5-0.12j];w = conj(z')


w = 1.0000 7.0000 + 4.0000i -15.6000 3.5000 - 0.1200i



Accessing Elements of Vectors – Subscript Notation

Consider the row vector

b = [b1 b2 b3 … bn]

Access to one or more elements of this vector is accomplished using subscript notation in which the subscript refers to the location in the vector as follows:

b(1) b1

…

b(k) bk

...

b(n) bn % or b(end)



Example –

x = [-2, 1, 3, 5, 7, 9, 10];b = x(4)c = x(6)xlast = x(end)

When executed, we obtain

b = 5c = 9xlast = 10



Accessing Elements of Vectors Using Subscript Colon Notation

The subscript colon notation is a shorthand method of accessing a group of elements.

For a vector b with n elements, one can access a group of them using the notation

b(k:d:m)

where 1 k < m n, k and m are positive integers and, in this case, d is a positive integer.

Thus, if one has 10 elements in a vector b, the third through seventh elements are selected by using

b(3:7)



Example –


z = [-2, 1, 3, 5, 7, 9, 10];z(2) = z(2)/2;z(3:4) = z(3:4)*3-1;z

When executed, we obtain

z = -2.00 0.50 8.00 14.00 7.00 9.00 10.00



Example –

Consider the script

y = [-1, 6, 15, -7, 31, 2, -4, -5];x = y(3:5)

whose execution creates the three-element vector

x = 15 -7 31



Example –

We shall show that there are several ways to create a vector x that is composed of the first two and the last two elements of a vector y. Thus,

y = [-1, 6, 15, -7, 31, 2, -4, -5];x = [y(1), y(2), y(7), y(8)]

or

y = [-1, 6, 15, -7, 31, 2, -4, -5];index = [1, 2, 7, 8]; % or [1:2, length(y)-1, length(y)] x = y(index)

or more compactly

y = [-1, 6, 15, -7, 31, 2, -4, -5];x = y([1, 2, 7, 8]) % or y([1:2, length(y)-1, length(y)])



Two useful functions: sort and find

sort: Sorts a vector y in ascending order (most negative to most positive) or descending order (most positive to most negative)

[ynew, indx] = sort(y, mode) % mode = 'ascend‘ or 'descend'

where ynew is the vector with the rearranged (sorted) elements of y and indx is a vector containing the original locations of the elements in y.

find: Determines the locations (not the values) of all the elements in a vector (or matrix) that satisfy a user-specified relational and/or logical condition or expression Expr

indxx = find(Expr)



Example –

y = [-1, 6, 15, -7, 31, 2, -4, -5];z = [10, 20, 30, 40, 50, 60, 70, 80];[ynew, indx] = sort(y, 'ascend')znew = z(indx)

when executed, gives

ynew = -7 -5 -4 -1 2 6 15 31indx = 4 8 7 1 6 2 3 5znew = 40 80 70 10 60 20 30 50



Example –

y = [-1, 6, 15, -7, 31, 2, -4, -5];indxx = find(y<=0)s = y(indxx)

Upon execution, we obtain

indxx = 1 4 7 8s = -1 -7 -4 -5



One of the great advantages of MATLAB’s implicit vector and matrix notation is that it provides the user with a compact way of performing a series of operations on an array of values.

Example –

The script

x = linspace(-pi, pi, 10);y = sin(x)

yields the following vector

y = -0.0000 -0.6428 -0.9848 -0.8660 -0.3420 0.3420 0.8660 0.9848 0.6428 0.0000



Minimum and Maximum Values of a Vector: min and max

To find the magnitude of the smallest element xmin and its location locmin in a vector, we use

[xmin, locmin] = min(x)

To find the magnitude of the largest element xmax and its location locmax in a vector, we use

[xmax, locmax] = max(x)



Example –

x = linspace(-pi, pi, 10);y = sin(x);[ymax, kmax] = max(y)[ymin, kmin] = min(y)

Upon execution, we find that

ymax = 0.9848kmax = 8ymin = -0.9848kmin =

3

y = -0.0000 -0.6428 -0.9848 -0.8660 -0.3420 0.3420 0.8660 0.9848 0.6428 0.0000



Example - Analysis of the elements of a vector

Consider 50 equally-spaced points of the following function

We shall

(a) determine the time at which the minimum positive value of f(t) occurs

(b) the average value of its negative values

The script is

( ) sin( ) 0 2f t t t



t = linspace(0, 2*pi, 50);f = sin(t);fAvgNeg = mean(f(find(f<0)))MinValuef = min(f(find(f>0)));tMinValuef = t(find(f==MinValuef))

Upon execution, we find that

fAvgNeg = -0.6237tMinValuef = 3.0775

0 1 2 3 4 5 6-1

-0.5

0

0.5

1



Creation of Matrices

The basic syntax to create a matrix is

A = [a11 a12 a13; a21 a22 a23; a31 a32 a33; a41 a42 a43]

where the semicolons are used to indicate the end of a row and the aij can be numbers, variable names, expressions, or strings.

Alternate ways are:

A = [a11 a12 a13; ... a21 a22 a23; ... a31 a32 a33; ... a41 a42 a43]

where the ellipses (...) are required to indicate that the expression continues on the next line.



For the second alternate way,

One can omit the ellipsis (…) and instead to use the Enter key to indicate the end of a row. In this case, the expression will look like

A = [a11 a12 a13 %<Enter> a21 a22 a23 %<Enter> a31 a32 a33 %<Enter> a41 a42 a43]



For a third alternate way,

Create four separate row vectors, each with the same number of columns, and then combine these vectors to form the matrix as follows

v1 = [ a11 a12 a13];v2 = [ a21 a22 a23];v3 = [ a31 a32 a33];v4 = [ a41 a42 a43]; A = [ v1; v2; v3; v4]

where the semicolons in the first four lines are used to suppress display to the command window.

The order of the matrix is determined by

[r, c] = size(A)



Transpose of a matrix with complex elements

If a matrix has elements that are complex numbers, then taking its transpose also converts the complex quantities to their complex conjugate values.


Z = [1+2j, 3+4j; 5+6j, 7+9j]W = Z'

the execution of which gives

Z = 1.0000 + 2.0000i 3.0000 + 4.0000i 5.0000 + 6.0000i 7.0000 + 9.0000iW = 1.0000 - 2.0000i 5.0000 - 6.0000i 3.0000 - 4.0000i 7.0000 - 9.0000i



Special Matrices

A matrix of all 1’s

ones(r, c) on(1:r,1:c) = 1 on = ones(2, 5) on = 1 1 1 1 1 1 1 1 1 1

A null matrix

zeros(r, c) zer(1:r,1:c) = 0 zer = zeros(3, 2) zer = 0 0 0 0 0 0



Diagonal matrix –

Create an (nn) diagonal matrix whose diagonal elements are a vector a of length n

diag(a)

a = [4, 9, 1];A = diag(a)A = 4 0 0 0 9 0 0 0 1

Extract the diagonal elements of a square matrix A

diag(A)

Ad = diag([11, 12, 13, 14; … 21, 22, 23, 24; … 31, 32, 33, 34; … 41, 42, 43, 44])Ad = 11 22 33 44



Identity matrix whose size is (nn)

eye(n) A = eye(3)A = 1 0 0 0 1 0 0 0 1



Accessing Matrix Elements

: means all elements of

column 2

: means all elements of

row 2

B = A(1:3,3:5)B = 7.0000 9.0000 11.0000 20.5000 20.7500 21.0000 1.0000 1.0000 1.0000

Matrix created with –

A = [3:2:11; … linspace(20, 21, 5); … ones(1, 5)];

3 5 7 9 11

20.0 20.25 20.5 20.75 21.0

1 1 1 1 1

A

A(1,1)

A(:,2) A(3,4)

A(1:3,3:5)

A(2,:)



In several of the following examples, we shall use

magic(n)

which creates an (nn) matrix in which the sum of the elements in each column and the sum of the elements of each row and the sum of the elements in each diagonal are equal.

For example,

magic(4)

creates

16 2 3 135 11 10 89 7 6 12

4 14 15 1



Example –

We shall set all the diagonal elements of magic(4) to zero; thus

Z = magic(4);Z = Z-diag(diag(Z))

which results in

Z= 0 2 3 13 5 0 10 8 9 7 0 12 4 14 15 0

diag(Z) = 16 11 6 1

diag(diag(Z)) = 16 0 0 0 0 11 0 0 0 0 6 0 0 0 0 1

magic(4) = 16 2 3 135 11 10 89 7 6 124 14 15 1



Example –

We shall replace all the diagonal elements of magic(4) with the value 5; thus,

Z = magic(4);Z = Z-diag(diag(Z))+5*eye(4)

which results in

Z = 5 2 3 13 5 5 10 8 9 7 5 12 4 14 15 5

magic(4) = 16 2 3 135 11 10 89 7 6 124 14 15 1



Use of find with Matrices

For a matrix,

[row, col] = find(Expr)

where row and col are column vectors of the locations in the matrix that satisfied the condition represented by Expr.

Let us find all the elements of magic(3) that are greater than 5. The script is

m = magic(3) [r, c] = find(m > 5);subscr = [r c] % Column augmentation




m = 8 1 6 3 5 7 4 9 2subscr = 1 1 3 2 1 3 2 3




M = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1minM = 4 2 3 1maxM = 16 14 15 13

Example –

M = magic(4)minM = min(M)maxM = max(M)

To find the maximum of all elements we use max twice:

M = magic(4);maxM = max(max(M))


maxM = 16

Minimum and Maximum Values in a Matrix

min and max operate on a column by column basis.



Example - Creation of a special matrix

We shall create the following (9×9) array

(Dashed lines have been added to enhance visual clarity.)

0 1 1 0 2 2 0 3 3

1 0 1 2 0 2 3 0 3

1 1 0 2 2 0 3 3 0

0 4 4 0 5 5 0 6 6

4 0 4 5 0 5 6 0 6

4 4 0 5 5 0 6 6 0

0 7 7 0 8 8 0 9 9

7 0 7 8 0 8 9 0 9

7 7 0 8 8 0 9 9 0

A



The script is

a = ones(3, 3)-eye(3);A = [a, 2*a, 3*a; 4*a, 5*a, 6*a; 7*a, 8*a, 9*a;]


A = 0 1 1 0 2 2 0 3 3 1 0 1 2 0 2 3 0 3 1 1 0 2 2 0 3 3 0 0 4 4 0 5 5 0 6 6 4 0 4 5 0 5 6 0 6 4 4 0 5 5 0 6 6 0 0 7 7 0 8 8 0 9 9 7 0 7 8 0 8 9 0 9 7 7 0 8 8 0 9 9 0



Example - Rearrangement of Sub Matrices of a Matrix

Consider the following (9×9) array

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81

A

1 2

4 3

(Dashed lines have been added to enhance visual clarity.)



For the four (33) sub matrices identified by the circled numbers 1 to 4, we shall perform a series of swaps of these sub matrices to produce the following array

61 62 63 4 5 6 55 56 57

70 71 72 13 14 15 64 65 66

79 80 81 22 23 24 73 74 75

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

7 8 9 58 59 60 1 2 3

16 17 18 67 68 69 10 11 12

25 26 27 76 77 78 19 20 21

A

3 4

2 1

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81

A

1 2

4 3



The script is

Matr = [1:9; 10:18; 19:27; 28:36; 37:45; ... 46:54; 55:63; 64:72; 73:81];

Temp1 = Matr(1:3, 1:3);Matr(1:3, 1:3) = Matr(7:9, 7:9);Matr(7:9, 7:9) = Temp1;Temp1 = Matr(1:3, 7:9);Matr(1:3, 7:9) = Matr(7:9, 1:3);Matr(7:9, 1:3) = Temp1;Matr




Matr = 61 62 63 4 5 6 55 56 57 70 71 72 13 14 15 64 65 66 79 80 81 22 23 24 73 74 75 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 7 8 9 58 59 60 1 2 3 16 17 18 67 68 69 10 11 12 25 26 27 76 77 78 19 20 21 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35 36

37 38 39 40 41 42 43 44 45

46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63

64 65 66 67 68 69 70 71 72

73 74 75 76 77 78 79 80 81

A

1 2

4 3



Manipulation of Arrays

Two functions that create matrices by replicating a specified number of times either a scalar, a column or row vector, or a matrix are

repmat

and

meshgrid

which uses repmat. The general form of repmat is

repmat(x, r, c)

where x is either a scalar, vector, or matrix, r is the total number of rows of x that will be replicated, and c is the total number of columns of x will be replicated.



Example –

We shall produce the following matrix three ways

W = 45.7200 45.7200 45.7200 45.7200 45.7200 45.7200 45.7200 45.7200 45.7200

Thus,

W = repmat(45.72, 3, 3) % Function

or

W = [45.72, 45.72, 45.72; … % Hard code 45.72, 45.72, 45.72; … 45.72, 45.72, 45.72]

or

W(1:3,1:3) = 45.72 % Subscript colon notation



Example –

Consider the vector

s = [a1 a2 a3 a4]

The expression

V = repmat(s, 3, 1)

creates the numerical equivalent of the matrix

1 2 3 4

1 2 3 4

1 2 3 4

=

a a a a

V a a a a

a a a aReplicated

row



whereas

repmat(s, 3, 2)

creates the numerical equivalent of the matrix

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

1 2 3 4 1 2 3 4

=

a a a a a a a a

V a a a a a a a a

a a a a a a a a

Replicated ‘column’

Replicated 1st row



On the other hand

V = repmat(s', 1, 3)

gives the numerical equivalent of the matrix

1 1 1

2 2 2

3 3 3

4 4 4

=

a a a

a a aV

a a a

a a a

Replicated 1st column



and

V = repmat(s', 2, 3)

gives the numerical equivalent of the matrix

1 1 1

2 2 2

3 3 3

4 4 4

1 1 1

2 2 2

3 3 3

4 4 4

=

a a a

a a a

a a a

a a aV

a a a

a a a

a a a

a a a



meshgrid

If we have two row vectors s and t, then

[U, V] = meshgrid(s, t)

gives the same result as that produced by the two commands

U = repmat(s, length(t), 1)

V = repmat(t', 1, length(s)) % Notice transpose

In either case, U and V are each matrices of order

(length(t)length(s))



Example –

If

s = [s1 s2 s3 s4] % (1×4) t = [t1 t2 t3] % (1×3)

then

[U, V] = meshgrid(s, t) % Sizes of U and V are (3×4)

produces the numerical equivalent of two (34) matrices

1 2 3 4

1 2 3 4

1 2 3 4

=

s s s s

U s s s s

s s s s

1 1 1 1

2 2 2 2

3 3 3 3

=

t t t t

V t t t t

t t t t



Example –

u = [1, 2, 3, 4];v = [5, 6, 7];[U, V] = meshgrid(u, v)


U = 1 2 3 4 1 2 3 4 1 2 3 4V = 5 5 5 5 6 6 6 6 7 7 7 7

However, u = [1, 2, 3, 4];v = [5, 6, 7];[V, U] = meshgrid(v, u)

gives

V = 5 6 7 5 6 7 5 6 7 5 6 7U = 1 1 1 2 2 2 3 3 3 4 4 4



There are two matrix manipulation functions that are useful in certain applications –

fliplr(A)

which flips the columns and

flipud(A)

which flips the rows.

Consider the (25) matrix

( )

11 12 13 14 15

21 22 23 24 25

= 2×5a a a a a

Aa a a a a

which is created with the statement

A = [a11 a12 a13 a14 a15; a21 a22 a23 a24 a25]



Then

15 14 13 12 11

25 24 23 22 21

21 22 23 24 25

11 12 13 14 15

( ) = (2×5)

( ) = (2×5)

a a a a aA

a a a a a

a a a a aA

a a a a a

fliplr

flipud

and

( ( ))

25 24 23 22 21

15 14 13 12 11

(2×5)a a a a a

Aa a a a a

flipud fliplr

These results are obtained with the colon notation. For example,

C = fliplr(A)

produces the same results as

C = A(:,length(A):-1:1)

( )

11 12 13 14 15

21 22 23 24 25

= 2×5a a a a a

Aa a a a a



Clarification of Notation –

Consider the following two (25) matrices A and B

11 12 13 14 15 11 12 13 14 15

21 22 23 24 25 21 22 23 24 25

a a a a a b b b b bA B

a a a a a b b b b b

Addition or subtraction: C = A B

11 11 12 12 13 13 14 14 15 15

21 21 22 22 23 23 24 24 25 25

± ± ± ± ±(2×5)

± ± ± ± ±

a b a b a b a b a bC

a b a b a b a b a b



Column augmentation: C = [A, B]

11 12 13 14 15 11 12 13 14 15

21 22 23 24 25 21 22 23 24 25

(2×10)a a a a a b b b b b

Ca a a a a b b b b b

Row augmentation: C = [A; B]

11 12 13 14 15

21 22 23 24 25

11 12 13 14 15

21 22 23 24 25

(4×5)

a a a a a

a a a a aC

b b b b b

b b b b b



Furthermore, if

x = [x1 x2 x3]y = [y1 y2 y3]

then either

Z = [x', y'] or Z = [x; y]' gives

whereas

Z = [x'; y'] yields

1 1

2 2

3 3

(2× 3)

x y

Z x y

x y

1

2

3

1

2

3

(6×1)

x

x

xZ

y

y

y



We illustrate these results with the following two vectors: a = [1, 2, 3] and b = [4, 5, 6]. The script is

x = [1, 2, 3];y = [4, 5, 6];Z1= [x', y']Z2= [x; y]'Z3= [x'; y']

Z1 = 1 4 2 5 3 6

Z3 = 1 2 3 4 5 6

Z2 = 1 4 2 5 3 6



Dot Operations

A means of performing, on matrices of the same order, arithmetic operations on an element-by-element basis.

Consider the following (34) matrices

11 12 13 14

21 22 23 24

31 32 33 34

x x x x

X x x x x

x x x x

11 12 13 14

21 22 23 24

31 32 33 34

m m m m

M m m m m

m m m m



11 11 12 12 13 13 14 14

21 21 22 22 23 23 24 24

31 31 32 32 33 33 34 34

* * * *

* * * *

* * * *m

x m x m x m x m

Z X.* M x m x m x m x m

x m x m x m x m

Then, the dot multiplication of X and M is

The dot division of X by M is

11 11 12 12 13 13 14 14

21 21 22 22 23 23 24 24

31 31 32 32 33 33 34 34

/m /m /m /m

/m /m /m /m

/m /m /m /md

x x x x

Z X./M x x x x

x x x x



11 11 12 12 13 13 14 14

21 21 22 22 23 23 24 24

31 31 32 32 33 33 34 34

^ ^ ^ ^

^ ^ ^ ^

^ ^ ^ ^e

x m x m x m x m

Z X.^ M x m x m x m x m

x m x m x m x m

The dot exponentiation of X and M is



Use of Dot Operations

Let a, b, c, d, and g each be a (32) matrix, and consider the expression

2

tand

bZ = a - g

c

Then the MATLAB expression is

Z = (tan(a)-g.*(b./c).^d).^2;

which is equivalent to

11 11 11 11 11 12 12 12 12 12

21 21 21 21 21 22 22 22 22 22

31 31 31 31 31 32 32 32 32 32

(tan( ) - * ( / )^ )^ 2 (tan( ) - * ( / )^ )^ 2

(tan( ) - * ( / )^ )^ 2 (tan( ) - * ( / )^ )^ 2

(tan( ) - * ( / )^ )^ 2 (tan( ) - * ( / )^ )^ 2

a g b c d a g b c d

Z a g b c d a g b c d

a g b c d a g b c d



Example –

Evaluate

1 1 1

1

sin( )a t b t cv e

t c

for 6 equally-spaced values of t in the interval 0 t 1 when a1 = 0.2, b1 = 0.9, and c1 = /6. The script is

a1 = 0.2; b1 = 0.9; c1 = pi/6;t = linspace(0, 1, 6);v = exp(-a1*t).*sin(b1*t+c1)./(t+c1)


v = 0.9549 0.8590 0.7726 0.6900 0.6097 0.5316



Example –

Create the elements of an (N×N) matrix H when each element is given by

1 , 1, 2,...

1mnh m n Nm n

The script to generate this array for N = 4 is

N = 4;mm = 1:N; nn = mm;[n, m] = meshgrid(nn, mm)h = 1./(m+n-1)

Upon execution, we get h = 1.0000 0.5000 0.3333 0.2500 0.5000 0.3333 0.2500 0.2000 0.3333 0.2500 0.2000 0.1667 0.2500 0.2000 0.1667 0.1429



We return to meshgrid and again consider the two vectors s = [s1 s2 s3 s4] and t = [t1 t2 t3]. Using the MATLAB expression

[U, V] = meshgrid(s, t) % U and V are (3×4) matrices

we obtained the matrices

If we perform the dot multiplication

Z = U.*V

then the execution of the program

1 2 3 4

1 2 3 4

1 2 3 4

=

s s s s

U s s s s

s s s s

1 1 1 1

2 2 2 2

3 3 3 3

=

t t t t

V t t t t

t t t t

and



s= [s1, s2, s3, s4];t = [t1, t2, t3];[U, V] = meshgrid(s, t) % (3×4) matricesZ = U.*V

results in the equivalent matrix

1 1 2 1 3 1 4 1

1 2 2 2 3 2 4 2

1 3 2 3 3 3 4 3

* * * *

* * * *

* * * *

s t s t s t s t

Z s t s t s t s t

s t s t s t s t

Thus, we have the product of all combinations of the elements of vectors s and t.



Summation and Cumulative Summation Functions

sum: When v = [v1, v2, …, vn], then

( )

1

( )v

kk

S v vlength

sum

where S a scalar.

If Z is a (34) matrix with elements zij, then

( )

3 3 3 3

1 2 3 4=1 =1 =1 =1

, , , (1×4)n n n nn n n n

S Z z z z zsum

To sum all the elements in an array, we use

( ( )) 3 3 3 3 4 3

1 2 3 4=1 =1 =1 =1 =1 =1

+ + + (1×1)n n n n nin n n n i n

S Z z z z z zsum sum



Example –

Evaluate

4

=1

m

m

z m

Thus,

m = 1:4;z = sum(m.^m)

which, when executed, gives

z = 288



cumsum: For a vector v composed of n elements vj, cumsum creates another vector of length n whose elements are

( )

1 2

=1 =1 =1

, , ... (1× )n

k k kk k k

y v v v v ncumsum

If W is (mn) matrix composed of elements wjk, then

( ) ( )

1 1 1

1 2=1 k =1 k =1

2 2

1 2=1 k =1

1=1 k =1

...

... ...

k k knk

k kk

m m

k knk

w w w

w wY W m×n

w w

cumsum



Example –

Consider the convergence of the series

N N=10

2=1

1

n

Sn

Thus,

S = cumsum(1./(1:10).^2)


S = 1.0000 1.2500 1.3611 1.4236 1.4636 1.4914 1.5118 1.5274 1.5398 1.5498



Example –

We shall evaluate the following series expression for the hyperbolic secant for N = 305 and for five equally spaced values of x from 0 x 2 and compare the results to the exact values.

4

( -1)/2

2 2=1,3,5

(-1)sech

+ 4

nN

n

nx

nπ xThus,

nn = 1:2:305; % (1153) xx = linspace(0, 2, 5); % (15)[x, n] = meshgrid(xx, nn); % (1535)s = 4*pi*sum(n.*(-1).^((n-1)/2)./…

((pi*n).^2+4*x.^2)); % (15)se = sech(xx); % (15)compare = [xx ' s' se' (100*(s-se)./se)'] % (52)




compare = 0.0000 1.0021 1.0000 0.2080 0.5000 0.8889 0.8868 0.2346 1.0000 0.6501 0.6481 0.3210 1.5000 0.4272 0.4251 0.4894

2.0000 0.2679 0.2658 0.7827 x approx sech(x) % error



Mathematical Operations with Matrices

Addition and Subtraction

If we have two matrices A and B, each of the order (mn), then

11 11 12 12 1n 1n

21 21 22 22

m1 m1 mn mn

± ± ... ±

± ±( × )

... ...

± ±

a b a b a b

a b a bA± B m n

a b a b

The MATLAB expression for matrix addition or subtraction is

W = A + B or W = A – B



Multiplication

If we have an (mk) matrix A and a (kn) matrix B, then

11 12 1

21 22

1

( )

n

m mn

c c c

c cC AB m n

c c

where

1

k

lp lj jpj

c a b

For the conditions stated, the MATLAB expression for matrix multiplication is

W = A*B



Note: The product of two matrices is defined only when the adjacent integers of their respective orders are equal; k in this case. In other words,

(mk)(kn) (mn)

where the notation indicates that we have summed the k terms.

1

k

lp lj jpj

c a b

Matrix Transpose

It can be shown that if C = AB, then its transpose is ( )C AB B A

If A is an identity matrix (A = I) and m = n, then

C IB BI B



Example –

We shall multiply the following two matrices and show numerically that the transpose can be obtained either of the two ways indicated previously.

11 1211 12 13

21 2221 22 23

31 32

A B

The script is

A = [11, 12, 13; 21, 22, 23];B = [11, 12; 21, 22; 31, 32];C = A*BCtran1 = C'Ctran2 = B'*A'




C = 776 812 1406 1472Ctran1 = 776 1406 812 1472Ctran2 = 776 1406 812 1472

A = [11, 12, 13; 21, 22, 23];B = [11, 12; 21, 22; 31, 32];C = A*BCtran1 = C'Ctran2 = B'*A'



Now consider the following series

( ) ( ) ( )k

j jj=1

w x, y f x g y

Suppose that we are interested in the value of w(x, y) over a range of values for x and y: x = x1, x2, …, xm and y = y1, y2, …, yn. Then one can consider

=1

( ) ( ) ( ) = 1,2, ..., = 1,2, ...,k

i j l i l jl

w x , y f x g y i m j n

as one element of a matrix W of order (mn) as follows. Let F be a matrix of order (mk)

( )

1 1 2 1 1

1 2 2 2

1

( ) ( ) ... ( )

( ) ( )

... ...

( ) ... ( )

k

m k m

f x f x f x

f x f xF m k

f x f x



and G be a matrix of order (kn)

( )

1 1 1 2 1

2 1 2 2

1

( ) ( ) ... ( )

( ) ( ) ...

... ...

( ) ... ( )

n

k k n

g y g y g y

g y g yG k n

g y g y

then

11 12 1

21 22

1

...

...( )

... ...

...

n

m mn

w w w

w wW FG m n

w w

where

1

( , ) ( ) ( )k

ij i j l i l jl

w w x y f x g y



We now consider three special cases of this general matrix multiplication:

1. The product of a row and a column vector

2. The product of a column and a row vector

3. The product of a row vector and a matrix

Thus, matrix multiplication performs the summation of the series at each combination of values for x and y.



Product of a Row Vector and Column Vector

Let a be the row vector

a = [a1 a2 … ak] (1k)

and b be the column vector

b = [b1 b2 … bk] (k1)

Then the product d = ab is the scalar

)

1

11 2

=1 =1

... = (1×1...

k k

k j j j jj j

k

b

bd ab a a a a b a b

b

(1k)(k1)

l =1 and p = 1

11 12 1

21 22

1

1

n

m mn

k

lp lj jpj

c c c

c cC AB

c c

c a b



This is called the dot product of two vectors.

The MATLAB notation for this operation is either

d = a*b

or

d = dot(a, b)



Product of a Column and Row Vector

Let a be an (m1) column vector and b a (1n) row vector. Then the product H = ba is

( )

1 1 1 1 2 1

2 2 1 2 21 2

1

...

......

... ... ...

...

n

n

m m m n

a a b a b a b

a a b a bH ba b b b m n

a a b a b

The elements of H, which are hij = biaj, are the individual products of all the combinations of the elements of b and a.

1

k

lp lj jpj

c a b

l = 1, 2, …, mp = 1, 2, …, n



Example - Polar to Cartesian Coordinates

Consider a vector of radial values

r = [r1 r2 … rm]

and a vector of angular values

= [1 2 … n]

then

*

1

21 2

1 1 1 2 1

2 1 2 2

1

cos( ) cos cos ... cos...

cos cos ... cos

cos cos

... ...

cos ... cos

n

m

n

m m n

r

rX r θ θ θ θ

r

r θ r θ r θ

r θ r θ

r θ r θ

r

y = rsin

x = rcos x

y



and

*

1

21 2

1 1 1 2 1

2 1 2 2

1

sin( ) sin sin ... sin...

sin sin ... sin

sin sin

... ...

sin ... sin

n

m

n

m m n

r

rY r θ θ θ θ

r

r θ r θ r θ

r θ r θ

r θ r θ

A typical MATLAB program would look like

r = [0:0.05:1]'; % (211)phi = 0:pi/20:2*pi; % (141)x = r*cos(phi); % (2141)y = r*sin(phi); % (2141)



Product of a Row Vector and a Matrix

Let B be an (mn) matrix and a a (1m) row vector. Then, the product g = aB is

( )

11 12 1

21 221 2

1

1 2=1 j=1 j=1

...

...... ...

...

... 1

n

m

m mn

m m m

j j j j j jnj

b b b

b bg aB a a a

b b

a b a b a b n

1

k

lp lj jpj

c a b

l = 1p = 1, 2, …, n



( ) =1

( )m

k kk

r x p h x

Now consider the series

Suppose that we are interested in the value of r(x) over a range of values x1, x2, …, xn. Then

is one element of a vector r of order (1n) as follows.

Let p be a vector of order (1m) with elements pk and V be the (mn) matrix

( )

1 1 1 2 1

2 1 2 2

1

( ) ( ) ... ( )

( ) ( )

... ...

( ) ( )

n

m m n

h x h x h x

h x h xV m n

h x h x

( ) =1

( ) = 1,2, ...,m

i k k ik

r x p h x i n



Then, r = pV gives

( ) 1 2

=1 =1 =1

( ) ( ) ... ( ) 1m m m

k k k k k k nk k k

r pV p h x p h x p h x n



Example - Summation of a Fourier Series

The Fourier series representation of a rectangular pulse of duration d and period T is given by

( )

=1

sin1 + 2 cos(2 )

K

k

kπd/Tdf πk

T kπd/T

where = t/T. Let us sum 150 terms of f() (K = 150) and plot it from 1/2 ≤ ≤ 1/2 when d/T = 0.25.

We see that

( ) ( )

( ) ( )

sin

cos(2 )

i

k

k i k i

r x f

kπd/Tp

kπd/T

h x h τ πk

( ) =1

( ) = 1,2, ...,m

i k k ik

r x p h x i n



The program is

k = 1:150; % (1150)tau = linspace(-0.5, 0.5, 100); % (1100)sk = sin(pi*k/4)./(pi*k/4); % (1150)cntau = cos(2*pi*k'*tau); % (150100)f = 0.25*(1+2*sk*cntau); % (1150) (150100) (1100)plot(tau, f)



Example -

31

1 cos( )( , ) 4 sin( )

Nn

n

nu e n

n

Assume that the length of is Nx and that the length of is Ne. Then, we note that

1 cos

sin3

(1 ) (1 )(1 ) (1 )[Not permissible](1 ) [Not permissible]

ex

n ne nn N NN N

N



1 cos

sin3

( 1)(1 )( 1)(1 )( ) (1 ) ( )

( )

eex

x

x

n ne nn N NN N

N NN N N

N N

In order to have the matrices compatible for multiplication, we take the transpose of vector n

1 cos

sin3

( )( )

[dot multiplication]

e

x

n ne nn N N

N N

From meshgrid(, f(n))



1 cos

sin ( )3

( )( ) ( )

e

x x

n ne n N Nx en N N

N N N N

Now we take the transpose and perform matrix multiplication

From the matrix multiplication of these two expressions, we are summing over n, which is what we set out to do.



n = (1:25)*pi; % (125)eta = 0:0.025:1; % (141)xi = 0:0.05:0.7; % (115)[X1, temp1] = meshgrid(xi, (1-cos(n))./n.^3); % (2515)temp2 = exp(-n'*xi); % (2515)Rnx = temp1.*temp2; % (2515)temp3 = sin(n'*eta); % (2541)u = 4*Rnx'*temp3; % (1541)mesh(eta, xi, u)

00.2

0.40.6

0.8

1

0

0.2

0.4

0.6

0.80

0.05

0.1

0.15

0.2

0.25

0.3

0.35



Determinants

A determinant of an array A of order (nn) is represented symbolically as

11 12 1

21 22

1

...

...

... ...

...

n

n nn

a a a

a aA

a a

For n = 2:

For n = 3:

11 22 12 21A a a a a

11 22 33 12 23 31 13 21 32 13 22 31

11 23 32 12 21 33

+ +

A a a a a a a a a a a a a

a a a a a a



The MATLAB expression for the determinant is

det(a)

Example –

If A is defined as 1 3

4 2A

then the determinant of A is obtained from

A = [1, 3; 4, 2];d = det(A)


d = -10



A class of problems, called eigenvalue problems, results in a determinant of the form

0A λB

where A and B are (nn) matrices and j, j = 1, 2, , …, n are the roots (called eigenvalues) of this equation.

One form of the MATLAB expression to obtain is

lambda = eig(A, B)



Example - Eigenvalues of a Spring-Mass System

The eigenvalues of a three-degree-of-freedom spring-mass system are obtained from the characteristic equation

2 0K - ω M

where

N/m

50 -30 0

-30 70 -40

0 -40 50

K kg

3 0 0

0 1.4 0

0 0 5

Mand

j jω = λand

0A λB



The program is

K = [50, -30, 0; -30, 70, -40; 0, -40, 50];M = diag([3, 1.4, 5]);w = sqrt(eig(K, M))


w = 1.6734 3.7772 7.7201

% rad/s

50 -30 0

-30 70 -40

0 -40 50

K

3 0 0

0 1.4 0

0 0 5

M



Matrix Inverse

The inverse of a square matrix A is an operation such that

-1 -1A A = AA = I

provided that A is not singular, that is, its determinant is not equal to zero (|A| 0).

The expression for obtaining the inverse of matrix A is either

inv(A)

or

A^-1

Note: 1/A A^-1; 1/A will cause the system to respond with an error message. However, 1./A is valid, but it is not the inverse.



Example - Inverse of Matrix

We shall determine the inverse of the magic matrix and the product of the inverse with the original matrix. The program is

invM = inv(magic(3))IdentMat = invM*magic(3)

Its execution gives

invM = 0.1472 -0.1444 0.0639 -0.0611 0.0222 0.1056 -0.0194 0.1889 -0.1028IdentMat = 1.0000 0 -0.0000 0 1.0000 0 0 0.0000 1.0000



Solution of a System of Equations

Consider the following system of n equations and n unknowns xk, k = 1, 2, …, n

11 1 12 2 1 1

21 1 22 2 2 2

1 1 2 2

+ + ... + =

+ + ... + =

...

+ + ... + =

n n

n n

n n nn n n

a x a x a x b

a x a x a x b

a x a x a x b

We rewrite this system of equations in matrix notation as follows:

Ax = b

where A is the following (nn) matrix



( )

11 12 1

21 22

1

...

...

... ...

...

n

n nn

a a a

a aA n n

a a

( ) ( )

1 1

2 21 and 1... ...

bn n

x b

x bx n b n

x

and

The symbolic solution is obtained by pre-multiplying both sides of the matrix equation by A-1. Thus,

since A-1A = I, the identity matrix, and Ix = x.

1 1

1

A Ax A b

x A b



The preferred expression for solving this system of equations is

x = A\b

where the backslash operator indicates matrix division and is referred to by MATLAB as left matrix divide. An equivalent way of solving a system of equations is by using

x = linsolve(A, b)



Example –

Consider the following system of equations

1 2 3

1 2 3

1 2 3

8 + + 6 = 7.5

3 + 5 + 7 = 4

4 + 9 + 2 = 12

x x x

x x x

x x x

which, in matrix notation, is

1

2

3

8 1 6 7.5

3 5 7 4

4 9 2 12

x

x

x

We shall determine xk and verify the results.



A = [8, 1, 6; 3, 5, 7; 4, 9, 2];b = [7.5, 4, 12]';x = A\b

z = A*x


x = 1.2931 0.8972 -0.6236

z = 7.5000 4.0000 12.0000

1

2

3

8 1 6 7.5

3 5 7 4

4 9 2 12

x

x

x



Example – Temperatures in a Slab

A thin square metal plate has a uniform temperature of 80 C on two opposite edges and a temperature of 120 C on the third edge and a temperature of 60 C on the remaining edge.

A mathematical procedure to approximate the temperature at six uniformly spaced interior points results in the following equations.

1 2 6

1 2 3 5

2 3 4

3 4 5

2 4 5 6

1 5 6

4 200

4 80

4 140

4 140

4 80

4 200

T T T

T T T T

T T T

T T T

T T T T

T T T



The temperatures are determined from the following script.

c = [4 -1 0 0 0 -1 -1 4 -1 0 -1 0 0 -1 4 -1 0 0 0 0 -1 4 -1 0 0 -1 0 -1 4 -1 -1 0 0 0 -1 4];d = [200, 80, 140, 140, 80, 200];T = linsolve(c, d')

The execution of this script gives T = 94.2857 82.8571 74.2857 74.2857 82.8571 94.2857

1 2 6

1 2 3 5

2 3 4

3 4 5

2 4 5 6

1 5 6

4 200

4 80

4 140

4 140

4 80

4 200

T T T

T T T T

T T T

T T T

T T T T

T T T

AN ENGINEER’S GUIDE TO MATLAB 3rd Edition CHAPTER 2 VECTORS and MATRICES

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Transcript of AN ENGINEER’S GUIDE TO MATLAB 3rd Edition CHAPTER 2 VECTORS and MATRICES