LECTURE 02

ARRAYS AND MATRICES

Shameer A Koya

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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 dimensions• Scalar - matrices with only one row and one column.

• Arrays can have numbers or letters

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Creating Matrices• In MATLAB, a vector is created by assigning the elements of

the vector to a variable. • This can be done in several ways depending on the source of

the information.• —Enter an explicit list of elements • —Load matrices from external data files • —Using built-in functions • —Using own functions in M-files

A matrix can be created in MATLAB by typing the elements (numbers) inside square brackets [ ]>> matrix = [1 2 3 ; 4 5 6 ; 7 8 9] 3

Examples• >> A = [2 -3 5; -1 4 5]

A = 2 -3 5 -1 4 5 >> x = [1 4 7] x = 1 4 7 >> x = [1; 4; 7] x = 1 4 7

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Note MATLAB displays row vector horizontally

Note MATLAB displays column vector vertically

Optional commas may be used between the elements

Type the semicolon (or press Enter) to move to the next row

>> 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

What if number of columns different?

>> B= [ 1:4; linspace(1,4,5) ]??? Error using ==> vertcatCAT arguments dimensions are not consistent.

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Four columns Five columns

Create vector with specified constant spacing between elements

The colon operator can be used to create a vector with constant spacing

x = m:q:n• m is first number• n is last number• q is difference between consecutive numbers >>x=[1:2:10] x = 1 3 5 7 9

If omit q, spacing is oney = m:n

>> y=1:5 y = 1 2 3 4 5

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Examples>> x = 1:2:13x = 1 3 5 7 9 11 13

>> >> 0.2:0.5:2.4 ans = 0.2000 0.7000 1.2000 1.7000 2.2000

>> 2:5 ans = 2 3 4 5

>> -2:3 ans = -2 -1 0 1 2 3

>> -3:3:10 ans = -3 0 3 6 9

>> 1.5:-0.5:-0.5 ans =

1.5000 1.0000 0.5000 0 -0.5000

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Non-integer spacing

No middle step value - default for the step is 1.

negative step is also possible

If the last numbers cannot be obtained by adding steps to first, then the last element in the vector will be the number that does not exceed last

Create vector with specified number of terms b/w first and last

v = linspace( xi, xf, n )• xi is first number• xf is last number• n is number of terms (= 100 if omitted)>>f= linspace(0, 10, 5)f = 0 2.5000 5.0000 7.5000 10.0000 5 elements, from 0 to 10

>>g= linspace(0, 10) g = Columns 1 through 10 0 0.1010 0.2020 0.3030 0.4040 ..................................................................Columns 91 through 100 9.0909 9.1919 9.2929 9.3939 9.4949 9.5960 9.6970 9.7980 9.8990 10.0000

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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

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Six elements

Decreasing elements

T I P

Built-in Functions to Generate Matrices

zeros(r,c) - makes matrix of r rows and c columns, all with zeros

ones(r,c) - makes matrix of r rows and c columns, all with ones

rand(r,c) - makes matrix of r rows and c columns, with random numbers

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

magic(n) - makes a special square matrix of n rows and c columns, called Durer’s matrix 10

Examples>> a=zeros(3,4)a = 0 0 0 0 0 0 0 0 0 0 0 0>> b=ones(4,3)b = 1 1 1 1 1 1 1 1 1 1 1 1>> c = rand(2,3)c =0.8147 0.1270 0.63240.9058 0.9134 0.0975

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>> d=eye(4)

d = 1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

>> e=magic(4)

e = 16 2 3 13

5 11 10 8

9 7 6 12

4 14 15 1

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

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T I P

Functions to Handle Matrices and Arrays• length( ) - number of elements in a array. number of columns in a matrix• >>length(y)• ans =• 3• >>length(x)• ans =• 3• • sum( ) – Sum of elements in a array. Sum of elements in a column of a

matrix.• >>sum(x)• ans =• 6• >>sum(y)• ans =• 5 7 9

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x = [1 2 3];y= [1 2 3; 4 5 6];

Functions to Handle Matrices and Arrays‘ – Transpose of a matrix. Convert a row vector to column vector

>>y'ans = 1 4 2 5 3 6>> x'ans = 1 2 3 diag( ) – diagonal elements of matrix. crate a matrix with elements of a vector,if the argument is a vector)>>diag(y)ans = 1 5>>diag(x)ans = 1 0 0 0 2 0 0 0 3

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x = [1 2 3];y= [1 2 3; 4 5 6];

Functions to Handle Matrices and Arrays• size( ) – size (dimensions) of matrix

>>size(y) ans = 2 3 (2 rows, 3 columns)

•det( ) – determinant of a matrix. (matrix must be square to get determinant)

• >> z= [1 2 3; 4 5 6; 7 8 0];• >> det (z)• ans =• 27• • inv( ) – inverse of a square matrix• >> inv(z)• ans =• -1.7778 0.8889 -0.1111• 1.5556 -0.7778 0.2222• -0.1111 0.2222 -0.1111

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x = [1 2 3];y= [1 2 3; 4 5 6];

Subscripts and Extracting a Sub-MatrixCan access (read from or write to) elements in array (vector or matrix) individually or in groups• Useful for changing subset of elements• Useful for making new variable from subset of elements

• Address (index) of element is its position in the array• x(n) – nth element in the vector• x(i, j)- element in ith row, jth column

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Examples>> A=[35 46 78 23 5 14 81 3 55]A = 35 46 78 23 5 14 81 3 5>> A(4)ans = 23>> A(6)=27A = 35 46 78 23 5 27 81 3 5>> A(2)+A(8)ans = 49>> A(5)^A(8)+sqrt(A(7))ans = 134 17

>> MAT=[3 11 6 5; 4 7 10 2; 13 9 0 8]

MAT = 3 11 6 5 4 7 10 2 13 9 0 8>> MAT(3,1)ans = 13>> MAT(3,1)=20MAT = 3 11 6 5 4 7 10 2 20 9 0 8>> MAT(2,4)-MAT(1,2)ans = -9 18

Element in row 3 and column 1

Assign new value to element in row 3 and column 1

Only this element changed

Column 1

Row 3

Using a Colon : in Addressing ArraysThe colon : lets you address a range of elements• Vector (row or column)• x(:) - all elements• x(m:n) - elements m through n

• Matrix• A(:,n) - all rows of column n• A(m,:) - all columns of row m• A(:,m:n) - all rows of columns m through n

• A(m:n,:) - all columns of rows m through n

• A(m:n,p:q) - columns p through q of rows m through n

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• >> A(3,:) % extract the 3rd row • ans =

7 8 9• >> A(:,2) % extract the 2nd

column• ans =

258

• >> A([1,3],1:2) % extract 1st and 2nd elements of 1st and 3rd rows• ans =

1 27 8

Adding And Deleting Elements• Indexing can be used to add and delete

elements from a matrix.• >> A(5,2) = 5 % assign 5 to the position (5,2); the

uninitialized elements become zeros

• >> A(4,:) = [2, 1, 2]; % assign vector [2, 1, 2] to the 4th row of A

• >> A(5,[1,3]) = [4, 4]; % assign: A(5,1) = 4 and A(5,3) = 4• A(4,:) = [ ] - will delete 4th row

• A(:, 3) = [ ] – will delete 3rd column

• A(1,2) = [ ] – error• Can’t delete single element in a row or column.

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A = 1 2 34 5 87 8 90 0 00 5 0

A(2:2:6) = [ ] ans = 1 7 5 3 6 9……………. How?

More Examplesv = 4 7 10 13 16 19 22 25 28 31 34>> u=v([3, 5, 7:10])u = 10 16 22 25 28 31

>> u=v([3 5 7:10])u = 10 16 22 25 28 31DF = 1 2 3 4>> DF(5:7)=10:5:20DF = 1 2 3 4 10 15 20 >> DF(10)=50DF = 1 2 3 4 10 15 20 0 0 50

>> A=[10:-1:4; ones(1,7); 2:2:14; zeros(1,7)]A = 10 9 8 7 6 5 4 1 1 1 1 1 1 1 2 4 6 8 10 12 14 0 0 0 0 0 0 0

>> B=A([1 3],[1 3 5:7])B = 10 8 6 5 4 2 6 10 12 14

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Appending vectors and matrices• Can only append row vectors to row vectors and column vectors to

column vectors• If r1 and r2 are any row vectors, r3 = [r1 r2] is a row vector whose left part is r1 and right part is r2

• If c1 and c2 are any column vectors, c3 = [c1; c2] is a column vector whose top part is c1 and bottom part is c2

• If appending one matrix to right side of other matrix, both must have same number of rows

• If appending one matrix to bottom of other matrix, both must have same number of columns

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Examplesr = [r1,r2]r = 1 2 3 4 5 6 7 8c = [c1; c2]

>> Z=[A B]Z = 1 2 3 7 8 4 5 6 9 10>> Z=[A; C]Z = 1 2 3 4 5 6 1 0 0 0 1 0 0 0 1>> Z=[A; B]??? Error using ==> vertcatCAT arguments dimensions are not consistent.

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r1 = [1 2 3 4 ]r2 = [5 6 7 8 ]

c1 = [1; 2; 3; 4 ]c2 = [5; 6; 7; 8 ]

A = 1 2 3 4 5 6B = 7 8 9 10C = 1 0 0 0 1 0 0 0 1

c = 1 2 3 4 5 6 7 8

Operations on Matrices• Previously dealt with scalars (single numbers). Will now

work with arrays, which in general have more than one number

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Addition and SubtractionWhen adding/ subtracting two arrays A and B, MATLAB adds/subtracts the corresponding elements (element wise addition/subtraction)When add/subtract a scalar to an array, MATLAB adds / subtracts the scalar to every element of the array

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Matrix MultiplicationThere are two ways of multiplying matrices – matrix multiplication and elementwise multiplicationMATRIX MULTIPLICATION• MATLAB denotes this with asterisk (*)• Number of columns in left matrix must be same as number of rows

in right matrix

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Matrix multiplication on two vectors• They must both be the

same size• One must be a row vector

and the other a column vector• If the row vector is on the

left, the product is a scalar• If the row vector is on the

right, the product is a square matrix whose side is the same size as the vectors 27

>> h = [ 2 4 6 ]h = 2 4 6>> v = [ -1 0 1 ]'v = -1 0 1>> h * vans = 4>> v * hans = -2 -4 -6 0 0 0 2 4 6

Array DivisionLeft division, \:Left division is one of MATLAB's two kinds of array division• Used to solve the matrix equation AX=B • A is a square matrix, X, B are column vectors• Solution is X = A-1B

In MATLAB, solve by using left division operator (\), i.e.,>> X = A \ B

When solving set of linear equations, use left division, not inverse, i.e., use X=A\B not X=inv(A)*BLeft division is• 2-3 times faster• Often produces smaller error than inv()• Sometimes inv()can produce erroneous results

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Right division, /:Right division is the other kind of MATLAB's array division• Used to solve the matrix equation XC=D • C is a square matrix, X, D are row vectors• Solution is X = D·C-1

In MATLAB, solve by using right division operator (/), i.e.,>> X = D / C

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Element-by-Element Operations• Addition and subtraction of arrays is always elementwise• Multiplication, division, exponentiation of arrays can be

elementwise• Both arrays must be same dimension

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ExampleELEMENTWISE MULTIPLICATION>> A = [1 2; 3 4];>> B = [0 1/2; 1 -1/2];>> C = [1 0]’;>> D = A .* BD =

0 13 -2

>> A .* C??? Error using ==> timesMatrix dimensions must agree.

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Built-in Functions for Analyzing Arrays• mean(v) – mean (average)• max(v) – maximum value, optionally with index of maximum• min(v) – minimum value, optionally with index of minimum• sum(v) – sum• sort(v) – elements sorted into ascending order• median(v) – median• std(v) – standard deviation• dot(v,w) – dot (inner product); v, w both vectors of same

size but any dimension• cross(v,w) – cross product; v, w must both have three

elements but any dimension

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Multi-dimensional Array

•Arrays can have more than two dimensions•A= [ 1 2; 3 4]•You can add the 3rd dimension by•A(:,:,2) = [ 5 6; 7 8]