Operations on Matrices and Arrays - MATLAB

Arithmetic operations of scalars are discussed in previous post. Now we will discuss about operations of matrices. All the basic operations on matrices, addition, subtraction, multiplication, division, and exponentiation, can be done in MATLAB. Operations other than addition and subtraction are done in two ways, one way, uses the standard symbols (*, /, and ^) where normal matrix operations are performed and the second way, which is called element-by-element operations, use a period typed in front of the standard operation symbol. There is an additional Matlab operator for matrices called left division operators (.\ or \).

Addition and Subtraction

Matrix addition can be accomplished only if the matrices to be added have the same dimensions for rows and columns.  If A and B are two matrices of the same size, then the sum A+B is the matrix obtained by adding the corresponding entries in A and B.

If A and B are matrices with the same size, then the difference between A and B denoted A-B is the matrix defined by subtracting the corresponding entries in B from A.

A scalar may be added to a matrix of any dimension.  When add/subtract a scalar to an array, MATLAB adds / subtracts the scalar to every element of the array. If A is a matrix, the expression A+n is evaluated by adding n to each element of A.

Examples:

 >> A = [5 1; 0 9];

>> B = [2 –2; 1 1];

>> A + B

ans =

     7    –1

     1    10

>> A – B

ans =

     3     3

    –1     8 

Matrix Multiplication

The multiplication operation * is executed by MATLAB according to the rules of linear algebra. This means that if A and B are two matrices, the operation A*B can be carried out only if the number of columns in matrix A is equal to the number of rows in matrix B. Such matrices are said to be conformable. Remember that the matrix multiplication is not commutative, i.e., A*B≠B*A (order of multiplication is important). A matrix of any dimension can be multiplied with a scalar. Each element in the matrix is multiplied by the scalar.

There are two ways of multiplying matrices – matrix multiplication and element wise multiplication. For element wise multiplication both matrices must be of same dimension.

Examples

>> A = [1 2; 3 4]

>> B = [0 1/2; 1 -1/2];

>> C = [1 0];

Matrix Multiplication

>> A*B

ans =

     0    16

     2    36

>> B*A

ans =

    14    20

    17    22

>> A*C

??? Error using ==> mtimes

Inner matrix dimensions must agree.

Element wise Multiplication

>> D = A .* B

D =

   0  1

   3 -2

>> A .* C

??? Error using ==> times

Matrix dimensions must agree.

Multiplication with scalar

>> A = [–2 2; 4 1];

>> C = 2*A

C =

    –4     4

     8     2 

Multiplication of two vectors

For multiplication of vectors, they must both be of 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

Example:

>> h = [ 2 4 6 ]

h =

     2     4     6

>> v = [ -1 0 1 ]'

v =

    -1

     0

     1

>> h * v

ans =

     4

>> v * h

ans =

    -2    -4    -6

     0     0     0

     2     4     6

dot product and cross product of vectors

Consider two vectors, X=[x1 x2] and Y = [y1 y2]. The inner product or dot product is defined as X.Y= x1* y1 +x2*y2 = |X|*|Y|*cos( ø). Another important operation involving vectors is the cross product. It is defined as XxY= |X|*|Y|*sin( ø).  In MATLAB, the dot product of two vectors A, B can be calculated using the dot(A,B) command. Cross(A,B) is used to compute the cross product.

>> a = [1;4;7]; b = [2;–1;5];

>> c = dot(a,b)

c =

    33

The dot product can be used to calculate the magnitude of a vector. All that needs to be done is to pass the same vector to both arguments. Consider the vector:

>> J = [0; 3; 4];

Calling dot we obtain:

>> dot(J,J)

ans =

    25

>> A = [1 2 3]; B = [2 3 4];

>> C = cross(A,B)

C =

    –1     2    –1

Matrix Division

The division operation is also associated with the rules of linear algebra. There are two types of division in Matlab - Right division and Left division. Right division performs the normal algebraic division of matrices. It can be used for solving the linear equations of the form XA=B. The solution of the equation is given by X=B/A.

Left division, \, is one of MATLAB's two kinds of array division. This type of division is used to solve the matrix equation AX=B,  A is a square matrix, X and B are column vectors. Solution of the system of equation is given by X = A \ B (Equivalent to inv(A)*B).

Left division is

• 2-3 times faster

• Often produces smaller error than inv()

• Sometimes inv()can produce erroneous results

Example

>> A = [1 2; 3 4];

>> B = [0 1/2; 1 -1/2];

>> A/B

ans =

     5     1

    11     3

>> A\B

ans =

    1.0000   -1.5000

   -0.5000    1.0000

>> A./B

ans =

   Inf     4

     3    -8

>> A.\B

ans =

         0         0.2500

    0.3333   -0.1250

Vector division

>> A = [2 4 6 8]; B = [2 2 3 1];

>> A/B

ans =

    2.1111

>> C = A./B

C =

     1     2     2     8

>> C = A.\B

C =

    1.0000    0.5000    0.5000    0.1250

Power of a Matrix

^ Operator is used to find the power of a matrix. The element wise operation also can be performed.

>> B = [2 4; –1 6]

B =

     2     4

    –1     6

>> B^2

ans =

     0    32

    -8    32

>> B.^2

ans =

     4    16

     1    36

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Indexing and Extracting a Sub-Matrix - MATLAB

Array Indexing

Each element in a array (matrix) has an address called index. Index of an element is its position in the array. The index for nth element in a vector x, is x(n). There are two ways to index a particular element in a matrix. The most common way is to specify row and column subscripts, such as, the element of row i and column j of the matrix A is denoted by A(i,j). The first index is the row number and the second index is the column number. For example, A(1,3) is an element of first row and third column. Another less common, but sometimes useful, is to use a single subscript, A(n), that traverses down each column in order.

We can access elements in array individually or in groups:

• Useful for changing subset of elements

• Useful for making new variable from subset of elements

Examples

Vector:

A = [0 1 2 3 4 5 6 7 8  9]

>> A(4)

ans = 3

>> A(6)=7

A = 0 1 2 3 4 7 6 7 8  9

>> A(3)+A(7)

ans = 8

>> A(4)^A(2)+sqrt(A(5))

ans = 13

Matrix:

B =  1    2     6     5

       4     3    7     2

      3     9     0     8

>> B(3,1)                              Element in row 3 and column 1

ans = 3

>> B(3,1)=0                         Assign new value to element in row 3 and column 1

B = 3    11     6     5

      4     7    10     2

     0     9     0     8

>> B(2,3) - B(1,2)

ans = 5

Using a Colon : in Addressing Arrays

The colon : lets you address a range of elements. The colon operator can also be used to pick out a certain row or column. For example, the statement A(m:n,k:l) specifies rows m to n and column k to l. Subscript expressions refer to portions of a matrix.

Vector (row or column) :

• x(:) - all elements
• x(m:n) - elements m through n
• x(end) – the last element

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,k:l) - columns k through l of rows m through n 
• A(end,:) - picks out the last row of A. The keyword end, denotes the last index in the specified dimension

Examples:

A = [1 2 3; 4 5 6; 7 8  9]

 >> A(3,:)     % extract the 3rd row

ans =

  7 8 9

 >> A(:,2)     % extract the 2nd column

ans =

2

5

8

>> A([1,3],1:2)    % extract 1^st and 2^nd  elements of 1^st and 3^rd row

ans =

1   2

7   8

To extract a submatrix B consisting of rows 2 and 3 and columns 1 and 2 of the matrix A, do the

following 

>> B = A([2 3],[1 2]) 

B  =

4  5

7  8

It is important to note that the colon operator (:) stands for all columns or all rows. To create a

vector version of matrix A, do the following

>> A(:)

ans =

1

2

3

4

5

6

7

8

9

>>A(end,:)

7  8  9

More Examples

v = 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 31

z = 1 2 3 4

>> z(5:7)=10:5:20

z = 1 2 3 4 10 15 20

>> z(10)=50

z = 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

Adding and Deleting Elements

Indexing can be used to add and delete elements from a matrix. An assignment operator can be used to add elements to add elements to an existing array. For example,

>> A(5,2) = 5  % assign 5 to the position (5,2)

A =

1  2  3

4  5  8

7  8  9

0  0  0

0  5  0

Two new rows are created and 2^nd element in the 5^th row is assigned number 5, and the uninitialized elements become zeros.

All elements in a row can be inserted as follows.

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

A =

1  2  3

4  5  8

7  8  9

2  1  2

0  5  0

Some selected elements in the array can be replaced as shown in the example below. Here the 1^st and 3^rd elements in the 5^th row is replaced by:

>> A(5,[1,3]) = [4, 4]       % assign: A(5,1) = 4 and A(5,3) = 4

A =

1  2  3

4  5  8

7  8  9

2  1  2

4  5  4

To delete a row or column of a matrix, use the empty vector operator, [ ].

Note: Can’t delete single element in a row or column.

>>A(4,:) = [ ]  % will delete 4 th row

A =

1  2  3

4  5  8

7  8  9

4  5  4

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

A =

1  2

4  5

7  8 

4  5 

A(1,2) = [ ]

??? error  -

A =

 1 2 3

 4 5 6

 7 8 9

A(2:2:6) = [ ]  % will remove the indexed elements in the matrix and the remaining elements will be displayed as a vector

ans = 1 7 5 3 6 9

The indexing can be used to recreate the deleted elements of an array. A(3,:) = [ ] will delete the third row of matrix A.  To restore the third row, we use a technique for creating a matrix.

>> A = [A(1,:);A(2,:);[7 8 9]]

A =

 1 2 3

 4 5 6

7 8 9

Matrix A is now restored to its original form.

Concatenating Matrices

Matrix concatenation is the process of joining one or more matrices to make a new matrix. The brackets [] operator discussed earlier in this section serves not only as a matrix constructor, but also as the MATLAB concatenation operator. The expression C = [A B] horizontally concatenates matrices A and B. The expression C = [A; B] vertically concatenates them.  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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Arrays and Matrices in MATLAB

—

MATLAB treats all variables as matrices. For our purposes a matrix can be thought of as an array, in fact, that is how it is stored. —Vectors are special forms of matrices and contain only one row or one column. —Scalars are matrices with only one row and one column.

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]

First open the [ bracket. Then type the elements in a row with a space or a comma between the elements and then enter the semicolon (or press Enter) to move to the next row. Finally close the ] bracket.

Example:

—>> 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

The Colon Operator and linspace Command

The colon operator can be used to create a vector with constant spacing (the difference between the elements is the same). The syntax is 

>>x= first:step:last

A vector with n elements that are linearly (equally) spaced in which the first element is xi and the last element is xf can be created by typing the linspace command. The syntax is 

>>x = linspace(xi, xf, n)

Example:

>> x=[1:2:10]

x =

     1     3     5     7     9   (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)

>> 2:5
ans =
    2   3   4   5 (If only the first and the last terms are typed then the default for the step is 1.)

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

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

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

>> 1.5:-0.5:-0.5     (negative step is also possible)
ans = 

     1.5000 1.0000 0.5000 0 -0.5000

>>f= linspace(0, 10, 5)

f =

         0    2.5000    5.0000    7.5000   10.0000     (5 elements, from 0 to 10)

>>f= linspace(0, 10)          (When the number of elements is omitted, the default is 100. 100 elements will be dispalyed)
f =
  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

Builtin Functions to Generate Matrices

Some special matrices can be created by using builtin functions. zeros (r, c), ones (r, c), eye (n) and rand(r, c) are the commands used to create matrix of r rows and c columns.

• zeros (r, c)  - create a matrix with all zeros
• ones (r, c)   - create a matrix with all ones
• eye (n)        - create a identity matrix
• rand(r, c)    - create a matrix with random numbers

Examples:

>> A = ones(3,2)
A =
     1     1
     1     1
     1     1
>> B = zeros(3,4)
B =
     0     0     0     0
     0     0     0     0
     0     0     0     0
>> C = rand(2,5)
C =
    0.8147    0.1270    0.6324    0.2785    0.9575
    0.9058    0.9134    0.0975    0.5469    0.9649
>> D = eye(3)
D =
     1     0     0
     0     1     0
     0     0     1
>> E = magic(4) - Durer’s matrix
E =
    16     2     3    13
     5    11    10     8
     9     7     6    12
     4    14    15     1

Functions to Handle Matrices and Arrays

x = [1 2 3];

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

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

‘   – 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

—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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