Manipulating MATLAB

Vector, Matrices

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Variables and ArraysWhat are variables?

You name the variables (as the programmer) and assign them numerical values.

You execute the assignment command to place the variable in the workspace memory (memory is part of hardware used for storing information).

You are allowed to use the variable in algebraic expressions, etc. once it is assigned.

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An array is MATLAB's basic data structure.

Can have any number of dimensions. Most common are:

vector - one dimension (a single row or column)

matrix - two or more dimensionsArrays can have numbers or letters.

(Arrays)

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In MATLAB, a variable is stored as an array of numbers. When appropriate, it is interpreted as a scalar, vector or matrix.

The size of an array is specified by the number of rows and the number of columns in the array, with the number of rows indicated first.

Variables as ArraysVariables as Arrays

scalar1 × 1

vectorn × 1 or 1 × n

matrixn × m

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• Scalars are 1×1 arrays.• They contain a single value, for example:

• X=3• Height=5.34• Width=10.09

Scalars

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VectorsA vector is a list of numbers expressed as a 1

dimensional array.A vector can be n×1 or 1×n.Columns are separated by commas (or spaces):x = [1, 2, 3] or x = [1 2 3]

Rows are separated by semicolons:y= [1; 2; 3]

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To create a row vector from known numbers, type variable name, then equal sign, then inside square brackets, numbers separated by spaces.

variable_name = [ n1, n2, n3 ]

>> yr = [1984 1986 1988 1990 1992 1994 1996]

yr =

1984 1986 1988 1990 1992 1994 1996

Note MATLAB displays row vector horizontally

Commas optional

(Creating a row vector)

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To create a column vector from known numbers

Method 1 - same as row vector but put semicolon after all but last number

variable_name = [ n1; n2; n3 ]

>> yr = [1984; 1986; 1988 ]

yr =

1984

1986

1988

Note: MATLAB displays column vector vertically

(Creating a Column Vector)

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Method 2 - same as row vector but put apostrophe (') after closing bracket.Apostrophe interchanges rows and

columns. Will come back on this later.variable_name = [ n1 n2 n3 ]'

>> yr = [1984 1986 1988 ]'

yr =

1984

1986

1988

(Creating a column vector)

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To create a vector with specified constant spacing between elements

variable_name = m:q:n

m is the first numbern is the last numberq is the difference between consecutive

numbersv = m:q:n

meansv = [ m m+q m+2q m+3q ... n ]

(Constant spacing vectors)

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If q is omitted, the spacing is 1v = m:n

meansv = [ m m+1 m+2 m+3 ... n ]

>> x = 1:2:13x = 1 3 5 7 9 11 13>> y = 1.5:0.1:2.1y = 1.5000 1.6000 1.7000 1.8000 1.9000 2.0000 2.1000

Non-integer spacing

(Constant spacing vectors)

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>> z = -3:7

z = -3 -2 -1 0 1 2 3 4 5 6 7

>> xa = 21:-3:6

xa = 21 18 15 12 9 6

>> fred = 7:-2:15

fred = Empty matrix: 1-by-0

Negative spacing

(Constant spacing vectors)

Impossible spacing

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1. How would you use linspace to set up spacings for planting seedlets in a garden row (30 seedlets, each row 2 meters long).

2. How about planning a 4285km road trip on 21 days. How long each leg would be?

(Quick linspace Exercises)

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To create a vector with specified number of terms between first and last

v = linspace( xi, xf, n )xi is first numberxf is last numbern is the number of terms (obviously must be a positivenumber) (= 100 if omitted)

(Fixed length vectors)

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>> va = linspace( 0, 8, 6 )

va = 0 1.6000 3.2000 4.8000 6.4000 8.0000

>> va = linspace( 30, 10, 11 )

va=30 28 26 24 22 20 18 16 14 12 10

m:q:n lets you directly specify spacing. linspace() lets you directly specify number of terms.

Six elements

Decreasing elements

T I P

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Indexing Into an Array allows you to change a value

Adding Elements

If you add an element outside the range of the original array, intermediate elements are added with a value of zero

2-D Matrices can also be entered by listing each row on a separate line C = [-12, 0, 0

12, 13, 40 1, -1, 90 0, 0, 2]

2-D Matrices

M = [12, 152, 6, 197, 32, -32, … 55, 82, 22, 10];

Use an ellipsis to continue a definition onto a new line

2-D Matrices

2-D matrix

These semicolons are optional

Define a matrix using other matrices as components

Or…

MatricesA matrix is a two

dimensional array of numbers.

For example, this is a 4×3 matrix:

mat1=[1.0,1.8,1.6; 3.0,-2.0,25.1; 0.0,-5.1,12.7; 2.3,0.3,-3.1]

11 22 33

11 1.01.0 1.81.8 1.61.6

22 3.03.0 -2.0-2.0 25.25.11

33 0.00.0 -5.1-5.1 12.12.77

44 2.32.3 0.30.3 -3.1-3.1

Columns

Row

s

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Indexed-location of numbers in an arrayEach item in an

array is located in the (row, column).

mat1(2,3)ans= 25.1000

11 22 33

11 1.01.0 1.81.8 1.61.6

22 3.03.0 -2.0-2.0 25.25.11

33 0.00.0 -5.1-5.1 12.12.77

44 2.32.3 0.30.3 -3.1-3.1

Columns

Row

s

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Enter the following into MATLAB:Scalar:

x=90

Vectors:y=[1, 0, 55]

z=[1 0 55]

Matrix: m= [ 55, 44, 33; 0, 2, 88]

Practice!

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Vector, Matrices

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Manipulating MATLAB

Defining (or assigning) arrays

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An array can be defined by typing in a list of numbers enclosed in square brackets:

Commas or spaces separate numbers.m=[12, 1, -33] or m=[12 1 -33]

Semicolons indicate a new row.

n=[2, 50, 20;1,1,2;0,-2,66]

m=m= 12 18 -3312 18 -33

n =n = 2 50 20 2 50 20 1 1 21 1 2 0 -2 660 -2 66

p =p = 12 18 -3312 18 -33 12 18 -3312 18 -33 2 50 202 50 20 2 50 202 50 20 1 1 21 1 2 1 1 21 1 2 0 -2 660 -2 66 0 -2 660 -2 66

Defining arrays continued

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You can define an array in terms of another array:r=[m;n]

p=[r,r]

r =r = 12 18 -3312 18 -33 2 50 20 2 50 20 1 1 21 1 2 0 -2 660 -2 66

We can access (read from or write to) elements in an array (vector or matrix) individually or in groups.

• Useful for changing a subset of elements.

• Useful for making a new variable from a subset of elements.

(Verctor/Matrix Addressing)

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(Accessing Vectors Elements 1)

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You can access any element by indexing the vector name with parentheses (indexing starts from 1, not from 0 as in C).

>> odd = [1:3:10] odd = 1 4 7 10

>> odd(3) ans = 7

Review…

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Index – a number used to identify elements in an arrayRetrieving a value from an array:

n(2,1) ans=1

You can change a value in an element in an array with indexing: n(3,2)=55

n =n = 2 50 20 2 50 20 1 1 21 1 2 0 -2 660 -2 66

n =n = 2 50 20 2 50 20 1 1 21 1 2 0 55 660 55 66

You can extend an array by defining a new element:

m=[12 1 -33] m(6)=9 m=[12 1 -33 0 0 9]

Notice how undefined values of the array are filled with zeros

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We can also access a range of elements (a subset of the original vector) by using the colon operator and giving a starting and ending index.

>> odd(2:4) ans = 4 7 10

Simple arithmetic between scalars and vectors is possible.

>> 10 + [1 2 3] ans = 11 12 13

(Range of Elements / Vector Arithmetic)

(Matrices)

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Create a two-dimensional matrix like this m = [ row 1 numbers; row 2 numbers; ... ; last row numbers ]

Each row separated by semicolon. All rows have same number of columns

>> a=[ 5 35 43; 4 76 81; 21 32 40]

a =

5 35 43

4 76 81

21 32 40

(Matrices)

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>> cd=6; e=3; h=4;

>> Mat=[e, cd*h, cos(pi/3);... h^2 sqrt(h*h/cd) 14]

Mat =

3.0000 24.0000 0.5000

16.0000 1.6330 14.0000

Comma is optional

(Matrices)

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Can also use m:p:n or linspace() to make rowsMake sure each row has same number of

columns>> A=[1:2:11; 0:5:25;... linspace(10,60,6); 67 2 43 68 4 13]

A =

1 3 5 7 9 11

0 5 10 15 20 25

10 20 30 40 50 60

67 2 43 68 4 13

(Matrices)

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What if the number of columns is different?

>> B= [ 1:4; linspace(1,4,5) ]

??? Error using ==> vertcat

CAT arguments dimensions are not consistent.

All arrys in the argument list must have the same number of columns.

Four columns

Five columns

(Special Matrix Commands)

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zeros(m,n) - makes a matrix of m rows and n columns, all with zeros.

ones(m,n) - makes a matrix of m rows and n columns, all with ones.

eye(n) - makes a square matrix of n rows and columns. Main diagonal (upper left to lower right) has ones, all other elements are zero.

Examples!

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zeros(2,5) ans = 0 0 0 0 0 0 0 0 0 0

ones(3,5) ans = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

eye(5) ans =

1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

Note: Placing a single number inside either function will return an n × n array. e.g. ones(5) will return a 5 × 5 array filled with ones.

>> zr=zeros(3,4)

zr = 0 0 0 0

0 0 0 0

0 0 0 0

>> ne=ones(4,3)

ne = 1 1 1

1 1 1

1 1 1

1 1 1

>> idn=eye(5)

idn = 1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

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To make a matrix filled with a particular number, multiply ones(m,n) by that number

>> z=100*ones(3,4)

z =

100 100 100 100

100 100 100 100

100 100 100 100

T I P

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(About variables in MATLAB)

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All variables are arraysScalar - array with only one elementVector - array with only one row or columnMatrix - array with multiple rows and columns

Assigning to variable specifies its dimensionDon't have to define variable size before assigning

to it, as you do in many programming languagesReassigning to variable changes its dimension to that

of assignment.

(The Transpose Operator)

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Transpose a variable by putting a single quote after it, e.g., x'In math, transpose usually denoted by

superscript "T", e.g., xTConverts a row vector to a column vector and vice-versa.

Switches rows and columns of a matrix, i.e., first row of original becomes first column of transposed, second row of original becomes second column of transposed, etc.

Continue…

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The transpose operator, an apostrophe, changes all of an array’s rows to columns and columns to rows.m = [12, 1, -33] m’

m= ans= 12 1 -33 12 1 -33

>> aa=[3 8 1]

aa = 3 8 1

>> bb=aa'

bb = 3

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