Matlab2011_c3.pdf

Chapter 3: Matrices, Calculations and Logical operations

3.1 Matrices

As mentioned in the earlier chapter, you can have lists of numbers as well as of letters.

These list of numbers can be one-dimensional, in which case they called a vector . When

they are two, three, or more-dimensional they are called a matrix .

mat1=[1 54 3; 2 1 5; 7 9 0; 0 1 0]

mat1 =

1 54 3

2 1 5

7 9 0

0 1 0

mat2=[1 54 3

2 1 5

7 9 0

0 1 0]

mat2 =

1 54 3

2 1 5

7 9 0

0 1 0

Take a look at mat1 and mat2 . As you can see there is more than one way of entering a

matrix. mat1 and mat2 were entered differently, but both have 4 rows and 3 columns.

When entering matrices a semi-colon is the equivalent of a new line.

You can find the size of matrices using the command size .

size(mat1)

ans =

4 3

For a two dimensional matrix the first value in size is the number of rows. The second

value of size is the number of columns.

Now try:

vect1=[1 2 4 6 3]

vect2=vect1'

vect1 =

1 2 4 6 3

vect2 =

Vectors can be tall instead of long, ' (the little symbol below the double quote on your

keyboard) is a transpose , and allows you to swop rows and columns of a vector or a matrix.

Remember the command whos which will tell you the size of all your variables at once?

Use whos to look at the size of mat1 , mat2 , vect1 and vect2 .

whos

Name Size Bytes Class Attributes

M1 2x3 48 double

SPM5Path 1x34 68 char

V1 1x4 32 double

V2 1x4 32 double

ans 1x2 16 double

mat 3x5 120 double

mat1 4x3 96 double

mat2 4x3 96 double

vect1 1x5 40 double

vect2 5x1 40 double

3.2 Addition and subtraction

You can perform various calculations on matrices and arrays. You can add a single number

(generally called a scalar ) to a vector.

vect1+3

ans =

4 5 7 9 6

You can subtract a scalar.

vect2-3

ans =

-2

-1

Scalar?

In physics a scalar is a value that only has magnitude (and not direction). In programming a scalar

is defined as a quantity that can only hold a single value at a time, i.e. a single number or

character. Generally the expression scalar tends to be used to refer to numbers rather than

characters.

Matrix dimensions must agree?

You get this error if you try to add vectors of inappropriate sizes. To successfully add two

vectors they must be the same orientation as well as the same length – i.e. you can't add a tall

thin vector to a short fat vector. The same goes for other operations (subtraction, point-wise

multiplication & division and matrix multiplication & division) and applies to matrices as

well as vectors.

When you get this error the first thing you should do is check the size of all the variables that

you are trying to manipulate, and try to work out why Matlab thinks they are mismatched in

size or shape. Often it’s the case that simply transposing one of the variables is all you need

to do. This is an important thing to understand, so play with a few examples of your own and

make sure you understand this.

You can add a vector onto itself

vect1+vect1

ans =

2 4 8 12 6

You can also add two vectors as long as they are the same size. You can’t add vect1 and

vect2 together since they are different sizes.

vect1+vect2

??? Error using ==> plus

Matrix dimensions must agree.

vect3= [1 2 3 4]

vect4=[ 1 3 5 2]';

vect3+vect4;

vect3 =

1 2 3 4

??? Error using ==> plus

Matrix dimensions must agree.

mat1=[1 2 3; 4 5 6];

mat2=[1 2; 3 4; 5 6];

mat1-mat2'

ans =

0 -1 -2

2 1 0

mat1-mat2

??? Error using ==> minus

Matrix dimensions must agree.

3.3 Scalar multiplication and division

Here you simply use the symbols * and /. You can a multiply a scalar with another scalar,

with a vector, or with a matrix.

2*3

ans =

2/3

ans =

0.6667

vect1*3

ans =

3 6 12 18 9

vect1/2

ans =

0.5000 1.0000 2.0000 3.0000 1.5000

mat1*0.5

ans =

0.5000 1.0000 1.5000

2.0000 2.5000 3.0000

3+1*4

ans =

3*4+1

ans =

Multiplication & division has priority over addition & subtraction. So the statements above

are the equivalent of 3+(1*4) and (3*4)+1. If you want to do addition or subtraction before

multiplication or division you need to use brackets.

(3+1)*4

ans =

3*(4+1)

ans =

Vector multiplication and

division

As you may or may not

remember from high school

math there are actually two

sorts of multiplication and

two sorts of division when

you are multiplying vectors

with each other (or matrices with each other). They are called point-wise and inner-product

multiplication respectively.

3.4 Point-wise vector multiplication and division

Point-wise (or element by element ) multiplication and division is the simplest. In Matlab

these are computed to using .* and ./ (notice the period character).

V1=[1 2 3 4];

V2=[2 3 4 5];

V1.*V2

ans =

2 6 12 20

V2.*V1

ans =

2 6 12 20

V1./V2

ans =

0.5000 0.6667 0.7500 0.8000

V2./V1

ans =

2.0000 1.5000 1.3333 1.2500

Each element in the first vector is multiplied by the corresponding element in the second

vector. As with addition and subtraction, the vectors must be the same shape. Note what

happens if we make one of the two vectors tall and thin (using the ' transpose ). We get

an error stating that we are trying to multiply two things of different shapes.

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