Engr/Math/Physics 25
Chp2 MATLAB
Arrays: Part-2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Learning Goals
 Learn to Construct 1D Row and Column
Vectors
 Create MULTI-Dimensional ARRAYS
and MATRICES
 Perform Arithmetic Operations on
Vectors and Arrays/Matrices
 Analyze Polynomial Functions
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Last Time
 Vector/Array Scalar
Multiplication by
Term-by-Term Scaling
>> r = [ 7 11 19];
>> v = 2*r
v =
14
22
38
 Vector/Array Addition &
Subtraction by Term-byTerm Operations
w rv
• “Tip-toTail” Geometry
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Scalar-Array Multiplication
 Multiplying an Array B by a scalar w produces
an Array whose elements are the elements of
B multiplied by w.
8   33.3 29.6
 9
3.7 



 12 14  44.4 51.8
 Using
MATLAB
>> B = [9,8;-12,14];
>> 3.7*B
ans =
33.3000
29.6000
-44.4000
51.8000
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Multiplication

Multiplication of TWO ARRAYS is not nearly
as straightforward as Scalar-Array Mult.

MATLAB uses TWO definitions for
Array-Array multiplication:
1. ARRAY Multiplication
– also called element-by-element multiplication
2. MATRIX Multiplication (see also MTH6)

DIVISION and EXPONENTIATION must also
be CAREFULLY defined when dealing with
operations between two arrays.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Element-by-Element Operations
Symbol
Operation
Form
Example
+
Scalar-array addition
A + b
[6,3]+2=[8,5]
-
Scalar-array subtraction
A – b
[8,3]-5=[3,-2]
+
Array addition
A + B
[6,5]+[4,8]=[10,13]
-
Array subtraction
A – B
[6,5]-[4,8]=[2,-3]
.*
Array multiplication
A.*B
[3,5].*[4,8]=[12,40]
./
Array right division
A./B
[2,5]./[4,8]=[2/4,5/8]
= [0.5,0.625]
.\
Array left division
A.\B
[2,5].\[4,8]=[2\4,5\8]
= [2,1.6]
A.^B
[3,5].^2=[3^2,5^2]
2.^[3,5]=[2^3,2^5]
[3,5].^[2,4]=[3^2,5^4]
.^
Array exponentiation
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Operations
 Array or Element-by-Element multiplication is
defined ONLY for arrays having the SAME
size. The definition of the product x.*y,
where x and y each have n×n elements:
x.*y = [x(1)y(1), x(2)y(2), ... , x(n)y(n)]
 if x and y are row vectors. For example, if
x
=
[2, 4, – 5],
y
=
[– 7, 3, – 8]
 then z = x.*y gives
z  2 7, 43,  5 8  14, 12, 40
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Operations cont
 If u and v are column vectors, the result of
u.*v is a column vector. The Transpose
operation z = (x’).*(y’) yields
 Note that x’ is a
 2 7    14
column vector with size




z   43    12 
3 × 1 and thus does not
have the same size as
 5 8  40 
y, whose size is 1 × 3
 Thus for the vectors x and y the operations
x’.*y and y.*x’ are NOT DEFINED in
MATLAB and will generate an error message.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Operations cont
 The array operations are performed between
the elements in corresponding locations in the
arrays. For example, the array multiplication
operation A.*B results in an array C that has
the same size as A and B and has the
elements cij = aij bij . For example, if
 11 5
A

  9 4
 7 8  Then C = A.*B
B

Yields
6
2


11 7  58  77 40
C








9
6
4
2

54
8

 

Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Operations cont
 The built-in MATLAB functions such as
sqrt(x) and exp(x) automatically operate
on array arguments to produce an array result
of the same size as the array argument x
• Thus these functions are said to be VECTORIZED
 Some Examples
>> r = [7 11 19];
>> h = sqrt(r)
h =
2.6458
3.3166
4.3589
Engineering/Math/Physics 25: Computational Methods
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>> u = [1,2,3];
>> f = exp(u)
f =
2.7183
7.3891
20.0855
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array Operations cont
 However, when multiplying or dividing these
functions, or when raising them to a power, we
must use element-by-element dot (.)
operations if the arguments are arrays.
 To Calc: z = (eu sinr)•cos2r, enter command
>> z = exp(u).*sin(r).*(cos(r)).^2
z =
1.0150
-0.0001
2.9427
 MATLAB returns an error message if the size
of r is not the same as the size of u. The
result z will have the same size as r and u.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array DIVISION
 The definition of array division is similar to the
definition of array multiplication except that the
elements of one array are divided by the
elements of the other array. Both arrays must
have the same size. The symbol for array right
division is ./
 Recall r = [ 7 11 19]
 then z = r./u gives
>> z = r./u
z =
7.0000
5.5000
Engineering/Math/Physics 25: Computational Methods
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u = [1,2,3]
z  7 1 11 2 19 3
 7 5.5 6.333
6.3333
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array-Array DIVISION cont.
 Consider
 24 20
A


9
4


 4 5
B

3
2


24  4 20 5  6 4
 Taking
C



C = A./B yields

9
3
4
2

3
2

 

A =
24
-9
20
4
-4
3
5
2
B =
Engineering/Math/Physics 25: Computational Methods
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>> A./B
ans =
-6
-3
4
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array EXPONENTIATION
 MATLAB enables us not only to raise arrays to
powers but also to raise scalars and arrays to
ARRAY powers.
 Use the .^ symbol to perform exponentiation
on an element-by-element basis
• if x = [3, 5, 8], then typing x.^3 produces the array
[33, 53, 83] = [27, 125, 512]
 We can also raise a scalar to an array power.
For example, if p = [2, 4, 5], then typing
3.^p produces the array [32, 34, 35] =
[9, 81, 243].
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Array to Array Power
>> A = [5 6 7; 8 9 8; 7 6 5]
A =
5
6
7
8
9
8
7
6
5
>> B = [-4 -3 -2; -1 0 1; 2 3 4]
B =
-4
-3
-2
-1
0
1
2
3
4
>> C = A.^B
C =
0.0016
0.0046
0.1250
1.0000
49.0000 216.0000
Engineering/Math/Physics 25: Computational Methods
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0.0204
8.0000
625.0000
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix-Matrix Multiplication
 Multiplication of MATRICES requires meeting
the CONFORMABILITY condition
 The conformability condition for multiplication
is that the COLUMN dimensions (k x m) of the
LEAD matrix A must be EQUAL to the ROW
dimension of the LAG matrix B (m x n)
 6  2
 If
 9 8


A  10 3 
B
 Then

 5 12

 4 7 
AB  C
C is 3x2 
Engineering/Math/Physics 25: Computational Methods
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BA  Error
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix-Mult. Mechanics
 Multiplication of A (k x m) and B (m x n)
CONFORMABLE Matrices produces a
Product Matrix C with Dimensions (k x n)
 The elements of C are the sum of the
products of like-index Row Elements from A,
and Column Elements from B; to whit
 6  2
69   2 5 68   212 64 24 
 9 8 


  75 116








C  10 3  

10
9

3

5
10
8

3
12
 
 


5
12


 4 7 
 49  7 5
48  712   1 116
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix-Vector Multiplication
 A Vector and Matrix May be Multiplied if they
meet the Conformability Critera: (1xm)*(mxp)
or (kxm)*(mx1)
• Given Vector-a, Matrix-B, and aB; Find Dims for all
c11
b11 b12 b13 
aB  a11 a12 
c

b21 b22 b23 
c12 c13   a11b11  a12b21 
a11b12  a12b22 
a11b13  a12b23 
 Then the Dims: a(1x2), B(2x3), c(1x3)
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt

 Summation Notation Digression
 Greek letter sigma (Σ, for sum) is another
convenient way of handling several terms or
variables – The Definition
q
sum   x p  x1 x2  x3   xq 2  xq 1  xq
p 1
 For the previous example
c
11
c12 c13   a11b11  a12b21 a11b12  a12b22 a11b13  a12b23 
a
p 1
b
1 p p1
Engineering/Math/Physics 25: Computational Methods
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2
2
2
a
p 1
b
1p p2
a
p 1
b
1 p p3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix Mult by Σ-Notation
 In General the
 c11
c
product of
21

Conformable
AB  C 

Matrices A & B when
A  k  m B  m  n

 ck 1
c12
c22

ck 2
 c1n 
 c2 n 
  

 ckn 
 Then Any Element, cij, of Matrix C for
• i = 1 to k (no. Rows)
j = 1 to n (no. Cols)
p m
p 7
p 1
p 1
cij   aipbpj e.g.; c53   a5 p bp 3
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix Mult Example
>> A = [3 1.7 -7; 8.1 -0.31 4.6; -1.2
2.3 0.73; 4 -.32 8; 7.7 9.9 -0.17]
A =
3.0000
8.1000
-1.2000
4.0000
7.7000
1.7000
-0.3100
2.3000
-0.3200
9.9000
-7.0000
4.6000
0.7300
8.0000
-0.1700
 A is Then 5x3
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix Mult Example cont
>> B = [0.67 -7.6;
4.4 .11; -7 -13]
>> C = A*B
C =
B =
0.6700
4.4000
-7.0000
-7.6000
0.1100
-13.0000
 B is Then 3x2
Engineering/Math/Physics 25: Computational Methods
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58.4900
68.3870
-28.1370 -121.3941
4.2060
-0.1170
-54.7280 -134.4352
49.9090 -55.2210
 Result, C, is Then
5x2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Matrix-Mult NOT Commutative
 Matrix multiplication is generally not
commutative. That is, AB ≠ BA
even if BA is conformable
• Consider an
Illustrative Example
1
A
3
2
0
&

4
6
 1
B

7
10  26 1 1  27  12 13 
AB  











3
0

4
6
3

1

4
7
24
25

 

01   13 02   14  3  4
BA  











6
1

7
3
6
2

7
4
27
40

 

Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Commutation Exceptions
 Two EXCEPTIONS to the NONcommutative
property are the NULL or ZERO matrix,
denoted by 0 and the IDENTITY, or UNITY,
matrix, denoted by I.
• The NULL matrix contains all ZEROS and is
NOT the same as the EMPTY matrix [ ],
which has NO elements.
 Commutation of the Null & Identity Matrices
0A  A0  0
IA  AI  A
 Strictly speaking 0 & I are always SQUARE
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Identity and Null Matrices
 Identity Matrix is a
1 0 0
square matrix and also it 1 0
0 1 0 etc.
or
.
is a diagonal matrix with 



0 1

1’s along the diagonal
0 0 1
• similar to scalar “1”
 Null Matrix is one in
which all elements are 0
• similar to scalar “0”
 Both are “idempotent”
Matrices: for A = 0 or I →
Engineering/Math/Physics 25: Computational Methods
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0 0 0 
0 0 0 


0 0 0
A  AT
and
A  A 2  A3 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
eye and zeros
 Use the eye(n)
command to
Form an nxn
Identity Matrix
 Use the
zeros(mxn)
to Form an mxn
0-Filled Matrix
• Strictly Speaking a
NULL Matrix is
SQUARE
Engineering/Math/Physics 25: Computational Methods
26
>> I = eye(5)
I =
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
>> Z24 = zeros(2,4)
Z24 =
0
0
0
0
0
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
0
0
PolyNomial Mult & Div
 Function conv(a,b) computes the product of
the two polynomials described by the coefficient
arrays a and b. The two polynomials need not be
the same degree. The result is the coefficient
array of the product polynomial.
 function [q,r] = deconv(num,den) produces
the result of dividing a numerator polynomial,
whose coefficient array is num, by a denominator
polynomial represented by the coefficient array
den. The quotient polynomial is given by the
coefficient array q, and the remainder polynomial
is given by the coefficient array r.
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
PolyNomial Mult Example
 Find the PRODUCT for
3
2


f x  2x  7 x  9x  6
>> f = [2 -7 9 -6];
>> g = [13,-5,3];
>> prod = conv(f,g)
prod =
26 -101 158 -144
2


g x  13x  5x  3
57
-18
prod x   26 x5  101x 4  158x3  144 x 2  57 x  18
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
PolyNomial Quotient Example
 Find the QUOTIENT
2 x3  7 x 2  9 x  6
quot 
13x 2  5 x  3
>> f = [2 -7 9 -6];
>> g = [13,-5,3];
>> [quot1,rem1] = deconv(f,g)
quot1 =
0.1538
-0.4793
rem1 =
0
0.0000
6.1420
quot x  0.1538x  0.4973
Engineering/Math/Physics 25: Computational Methods
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-4.5621
rem 6.142 x  4.5621
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
PolyNomial Roots
 The function roots(h) computes the roots of
a polynomial specified by the coefficient array
h. The result is a column vector that contains
the polynomial’s roots.
>> r = roots([2, 14, 20])
r =
2
2 x  14 x  20  0  x  5 x  2
-5
-2
>> rf = roots(f)
>> rg = roots(g)
rf =
rg =
2.0000
0.1923 + 0.4402i
0.7500 + 0.9682i
0.1923 - 0.4402i
0.7500 - 0.9682i

Engineering/Math/Physics 25: Computational Methods
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
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt

Plotting PolyNomials
 The function polyval(a,x)evaluates a
polynomial at specified values of its
independent variable x, which can be a matrix
or a vector. The polynomial’s coefficients of
descending powers are stored in the array a.
The result is the same size as x.
 Plot over (−4 ≤ x ≤ 7) the function
f x   2 x  7 x  9 x  6
3
Engineering/Math/Physics 25: Computational Methods
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2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
The Demo Plot
Engineering/Math/Physics 25: Computational Methods
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
All Done for Today
Cell-Arrays
&
Structure
Arrays
Engineering/Math/Physics 25: Computational Methods
33
 Not Covered in
Chapter 2
• §2.6 = Cell Arrays
• §2.7 = Structure
Arrays
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Engr/Math/Physics 25
Appendix
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering/Math/Physics 25: Computational Methods
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Chp2 Demo
x = linspace(-4,7,20);
p3 = [2 -7 9 -6];
y = polyval(p3,x);
plot(x,y, x,y, 'o', 'LineWidth', 2), grid,
xlabel('x'),...
ylabel('y = f(x)'), title('f(x) = 2x^3 - 7x^2 +
9x - 6')
f(x) = 2x 3 - 7x 2 + 9x - 6
500
400
300
y = f(x)
200
100
0
-100
-200
-300
-4
-2
Engineering/Math/Physics 25: Computational Methods
35
0
2
x
4
6
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt
Chp2 Demo
>> f = [2 -7 9 -6];
>> x = [-4:0.02:7];
>> fx =
polyval(f,x);
>>
plot(x,fx),xlabel('x
'),ylabel('f(x)'),
title('chp2 Demo'),
grid
Engineering/Math/Physics 25: Computational Methods
36
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-25_Arrays-2.ppt