introduction to matlab vector and matrix

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Introduction to MatlabVector and Matrix Operation

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Matlab basic operations

• MATLAB is based on matrix/vector mathematics

• Entering matrices

• Enter an explicit list of elements

• Load matrices from external data files

• Generate matrices using built-in functions

• Create vectors with the colon (:) operator

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Entering a Matrix in MATLAB

2 -3 5-1 4 6

A

• MATLAB Format:

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

A =

2 -3 5

-1 4 5

3

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Entering a Row Vector in MATLAB

[1 4 7]x

• MATLAB Format:

>> x = [1, 4, 7]

x =

1 4 7

4

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Entering a Column Vector in MATLAB

1

4

7

x

• MATLAB Format

>> x = [1; 4; 7]

x =

1

4

7

5

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Generate matrices using built-infunctions

• Functions such as zeros(), ones(), eye(), magic(), etc.

>> A=zeros(3)

 A =

0 0 0

0 0 0

0 0 0

>> B=ones(3,2)

B =

1 1

1 1

1 1 

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>> I=eye(4) (i.e., identity matrix)

I =

1 0 0 0

0 1 0 0

0 0 1 00 0 0 1

>> A=magic(4) (i.e., magic square)

A =

16 2 3 13

5 11 10 8

9 7 6 124 14 15 1

>>

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Generate Vectors with Colon (:) Operator

The colon operator uses the following rules to create regularly spaced

vectors:

 j:k is the same as [j,j+1,...,k]

 j:k is empty if j > k

 j:i:k is the same as [j,j+i,j+2i, ...,k]

 j:i:k is empty if i > 0 and j > k or if i < 0 and j < k

where i, j, and k are all scalars.

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>> c=0:5

c =

0 1 2 3 4 5

>> b=0:0.2:1

b =0 0.2000 0.4000 0.6000 0.8000 1.0000

>> d=8:-1:3

d =

8 7 6 5 4 3

>> e=8:2

e =

Empty matrix: 1-by-0 

Examples

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Extracting a Sub-Matrix

• A portion of a matrix can be extracted and stored in a smaller matrixby specifying the names of both matrices, the rows and columns. Thesyntax is:

sub_matrix = matrix ( r1 : r2 , c1 : c2 ) ;

where r1 and r2 specify the beginning and ending rows and c1 and c2 specify the beginning and ending columns to be extracted to make thenew matrix.

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MATLAB Matrices

• A column vector can beextracted from a matrix. Asan example we create amatrix below:

» matrix=[1,2,3;4,5,6;7,8,9]

matrix =

1 2 3

4 5 67 8 9

• Here we extract column 2 ofthe matrix and make acolumn vector:

» col_two=matrix( : , 2)

col_two =

2

5

8

11

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MATLAB Matrices

• A row vector can be extractedfrom a matrix. As an examplewe create a matrix below:

» matrix=[1,2,3;4,5,6;7,8,9]

matrix =

1 2 3

4 5 67 8 9

• Here we extract row 2 of thematrix and make a row vector.Note that the 2:2 specifies thesecond row and the 1:3specifies which columns of therow.

» rowvec=matrix(2 : 2 , 1 : 3)

rowvec =

4 5 6

12

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Matrices transpose

• a vector x = [1, 2, 5, 1]

x =

1 2 5 1

• transpose y = x’ y =

1

2

5

1

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Basic Permutation of Matrix in MATLAB

• sum, transpose, and diag

• Summation

We can use sum() function.

Examples, 

>> X=ones(1,5)

X =

1 1 1 1 1

>> sum(X)

ans =

5

>>

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>> A=magic(4)

 A =

16 2 3 13

5 11 10 8

9 7 6 124 14 15 1

>> sum(A)

ans =

34 34 34 34

>>

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Transpose

>> A=magic(4)

 A =

16 2 3 13

5 11 10 8

9 7 6 124 14 15 1

>> A'

ans =

16 5 9 42 11 7 14

3 10 6 15

13 8 12 1

>>

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Scalar - Matrix Addition

» a=3;

» b=[1, 2, 3;4, 5, 6]

b =

1 2 3

4 5 6

» c= b+a % Add a to each element of b

c =

4 5 6

7 8 9

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Scalar - Matrix Subtraction

» a=3;

» b=[1, 2, 3;4, 5, 6]

b =

1 2 34 5 6

» c = b - a %Subtract a from each element of b

c =

-2 -1 01 2 3

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Scalar - Matrix Multiplication

» a=3;

» b=[1, 2, 3; 4, 5, 6]

b =1 2 3

4 5 6

» c = a * b % Multiply each element of b by a

c =

3 6 9

12 15 18

19

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Scalar - Matrix Division

» a=3;

» b=[1, 2, 3; 4, 5, 6]

b =1 2 3

4 5 6

» c = b / a % Divide each element of b by a

c =

0.3333 0.6667 1.0000

1.3333 1.6667 2.0000

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Matrix Addition and Subtraction

•Matrix addition and subtraction with MATLAB are achieved in thesame manner as with scalars provided that the matrices have thesame size. Typical expressions are shown below.

>> C = A + B

>> D = A - B

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Matrix Multiplication

Matrix multiplication with MATLAB isachieved in the same manner as withscalars provided that the number of

columns of the first matrix is equal to thenumber of rows of the second matrix. Atypical expression is

>> E = A*B

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Matrix Multiplication

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 Array Multiplication

•There is another form of multiplication of matrices in which it isdesired to multiply corresponding elements in a fashion similar tothat of addition and subtraction. This operation arises frequently withMATLAB, and we will hereafter refer to the process as the arrayproduct to distinguish it from the standard matrix multiplication form.

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 Array Multiplication Continuation

•For the array product to be possible, the two matrices must have thesame size, as was the case for addition and subtraction. The resultingarray product will have the same size. If F represents the resultingmatrix, a given element of F, denoted by f ij   is determined by thecorresponding product from the two matrices as

ij ij ij f a b26

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MATLAB Array Multiplication

•To form an array product in MATLAB, a period must be placed afterthe first variable. The operation is commutative. The following twooperations produce the same result.

>> F=A.*B

>> F=B.*A

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MATLAB Array MultiplicationContinuation

If there are more than two matrices forwhich array multiplication is desired, the

periods should follow all but the last onein the expression; e. g., A.*B.*C in thecase of three matrices. Alternately,nesting can be used; e.g. (A.*B).*C for

the case of three matrices.

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MATLAB Array MultiplicationContinuation

•The array multiplication concept arises in any operation in which thecommand could be “confused” for a standard matrix operation. Forexample, suppose it is desired to form a matrix B from a matrix A byraising each element of A to the 3rd power, The MATLAB command is

>> B = A.^3

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Determinant of a Matrix

•The determinant of a square matrix in MATLAB is determined by thesimple command det(A). Thus, if a is to represent the determinant,we would type and enter

>> a = det(A)

•Note that a is a scalar (1 x 1 "matrix").

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Inverse Matrix

•The inverse of a square matrix in MATLAB is determined by the simplecommand inv(A). Thus, if B is to represent the inverse of A , thecommand would be

>> B = inv(A)

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Simultaneous Equation Solution

Ax = b

•MATLAB Format:

>> x = inv(A)*b

•Alternate MATLAB Format:

>> x = A\b

34

-1x = A b

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Example 3-1. Enter the matrices belowin MATLAB. They will be used in the

next several examples.

2 -3 5

-1 4 6

A

>> A = [2 -3 5; -1 4 6];

>> B = [2 1; 7 -4; 3 1];

35

2 1

7 -4

3 1

B

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Example 3-2. Determine the transposeof B and denote it as C.

>> C = B'

C =

2 7 3

1 -4 1

•The 3 x 2 matrix has been converted to a 2 x 3 matrix.

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Example 3-3. Determine the sum of Aand C and denote it as D.

>> D = A + C

D =

4 4 8

0 0 7

37

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Example 3-4. Determine the product of Aand B with A first.

>> A*B

ans =

-2 19

44 -11

38

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Example 3-5. Determine the product of Band A with B first.

>> B*A

ans =

3 -2 16

18 -37 11

5 -5 21

39

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Example 3-6. Determine the array productof A and C and denote it as E.

>> E = A.*C

E =

4 -21 15

-1 -16 6

40

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Example 3-7. Enter the matrix A. It will be usedin several examples.

1 2 -1

-1 1 3

 3 2 1

A

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

A =

1 2 -1

-1 1 3

3 2 1

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Example 3-7. Continuation. Determine thedeterminant of A and denote it as a.

>> a = det(A)

a =

20

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Example 3-8. Determine the inverse matrixof A and denote it as Ainv.

>> Ainv = inv(A)

Ainv =

-0.2500 -0.2000 0.3500

0.5000 0.2000 -0.1000

-0.2500 0.2000 0.1500

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Example 3-9. Use MATLAB to solvethe simultaneous equations below.

1 2 32 8 x x x

1 2 33 7 x x x

1 2 33 2 4 x x x

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Example 3-9. Continuation.

• Assume that A is still in memory.

>> b = [-8; 7; 4]

b =

-8

7

4

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Example 3-9. Continuation.

>> x = inv(A)*b

x =

2.0000

-3.0000

4.0000

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Example 3-9. Continuation.• Alternately,

>> x = A\b

x =

2.0000

-3.0000

4.0000