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MATLAB Workshop 12 - Matrices (Arrays)
Objectives : Learn about matrix properties in MATLAB, methods to create matrices, mathematical
functions with matrices, element-by-element matrix operations, and matrix algebra.
MATLAB Features :
vector/matrix variable properties
Property Comment
var_name
user chooses names
- name should represent meaning of variable ex: use radius rather than x or s for radius
- letters, digits (0-9), underscore ( _ ) allowed in name
- must start with a letter
- case sensitive ( Radius is different than radius )
- can be any length, but first 31 characters must be unique
value
all numerical values are
- double precision ~ 16 significant figures (not decimal places)
- and imaginary has both real and imaginary parts
a ±±±± bi (where i = ( − 1 ))
both a and b are double precision if b = 0, MATLAB only displays real part, a memory location MATLAB assigns - you do not need to worry about this
order of precedence rules for vector/matrix arithmetic operators
Order of Precedence Symbol Meaning
1 ( )^ group together ' (^) transpose (exchange rows and columns) 2 .^ (^) raise each element to indicated power ^ (^) multiply matrix by itself indicated number of times .* (^) element-by-element multiplication ./ (^) element-by-element right division .\ (^) element-by-element left division
- (^) multiply two matrices / (^) matrix right division
\ (^) matrix left division
colon ( : ) operator - a special vector/matrix operator
Notation Action Result a:c (^) creates vector with elements 1 apart [a a+1 a+2 ... c] a:b:c creates vector with elements b apart (^) [a a+b a+2b ... (≤≤≤≤c)] A(m,:) (^) selects m th^ row of A A(:,j) selects j th^ column of A
A(m:n,:) selects m th^ through nth^ rows of A A(:,j:k) selects j th^ through k th^ columns of A
A(m:n,j:k) selects m th^ through nth^ rows of A and j th^ through k th^ columns of A
matrix functions
number of rows and columns
[rows, cols] = size(x)
» Array2trans = Array2' Array2trans = 3 -4 12 12 -3 5
Array2 has been “transposed”, i.e., the columns and rows were interchanged so that the first column became the first row, etc.
(3) Extracting rows or columns from an array in MATLAB.
Enter the following at the Command Line prompt » row2_Array1 = Array1(2,:) row2_Array1 = 4 -3 6 The second row of Array1 has been extracted using the colon operator.
Enter the following at the Command Line prompt » col1_Array2 = Array2(:,1) col1_Array2 = 3
12 The first column of Array2 has been extracted using the colon operator.
Enter the following at the Command Line prompt » Array1sub = Array1(1:2,2:3) Array1sub = 9 - -3 6 A subset array consisting of elements from the 1st^ and 2nd^ rows, 2nd^ and 3rd^ columns of Array1 has been created.
(4) Determining the size of a matrix.
Enter the following at the Command Line prompt » size(Array1) ans = 3 3 size returns the number rows and columns in an array.
Enter the following at the Command Line prompt » [A2rows, A2cols] = size(Array2) A2rows = 3 A2cols = 2 We can assign the number of rows and columns in a matrix to variables for later use.
Enter the following at the Command Line prompt
- The colon operator is a powerful tool for extracting parts of an array for further use.
» size(row2_Array1) ans = 1 3 The size of a row vector is (1,N) for 1 row, and N elements. What do you think the size of a column vector is? Contrast size with length for a vector.
- Function and element-by-element operations with matrices in MATLAB
MATLAB processes mathematical functions and element-by-element operations for matrices in
the same manner as it does for vectors. That is, application of a function, e.g.
B = exp(A)
where A is an array will raise every element in A to the eth^ power to produce the corresponding elements
of array B. Likewise, the element-by-element operators, .^, .*, ./, and ., are used to process
matrices element-by-element.
(5) Functions of matrices in MATLAB.
Enter the following at the Command Line prompt » Amat = [1 10; 100 1000] Amat = 1 10 100 1000 » log10_Amat = log10(Amat) log10_Amat = 0 1. 2.0000 3. » » Bmat = [0 pi/6 pi/3; pi/2 2pi/3 5pi/6] Bmat = 0 0.5236 1. 1.5708 2.0944 2. » sin_Bmat = sin(Bmat) sin_Bmat = 0 0.5000 0. 1.0000 0.8660 0.
(6) Scalar-matrix operations in MATLAB.
Enter the following at the Command Line prompt » const = 5; » Cmat = Amat + const Cmat = 6 15 105 1005 The constant value 5 was added to each element of Cmat.
Enter the following at the Command Line prompt
- The size function returns the number of rows and columns in a matrix.
- Mathematical functions operate on a matrix element-by-element.
Cmat are displayed as 1.0e+003 *. This means that each number that follows should be multiplied by 10 3 in order to produce the proper element value. MATLAB automatically moves to scientific notation type display when necessary.
Contrast the preceding with » Hmat = Amat*log10_Amat Hmat = 1.0e+003 * 0.0200 0. 2.0000 3. Using the multiplication operator, , instead of the element-by-element multiplication operator, ., instructed MATLAB to follow the rules of matrix multiplication (outlined below) and produces an entirely different answer.
Matrices are characterized by their dimension. For simplicity, [ A ] MxN will denote a matrix with
M rows and N columns. Likewise, ai,j will denote the element value in the i
th
row, j
th
column of
[ A ] MxN. The row dimension will always come first and the column dimension second. The rules for
basic matrix algebra operations are as follows.
Matrix addition/subtraction. Two matrices can be added or subtracted if and only if they have the
same dimensions. The result is a matrix with the same dimensions. Thus,
[ C ] 3 x 2 =[ A ] 3 x 2 +[ B ] 3 x 2
is possible while
[ A ] 3 x 2 +[ D ] 2 x 3
is not. Addition of two matrices is done element-by-element as
c i , j = ai , j + bi , j
for each element in the matrices. Matrix subtraction is defined as
d i , j = ai , j − bi , j
Note that matrix addition is commutative and matrix subtraction is not commutative.
Matrix multiplication. Two matrices can be multiplied if and only if the number of columns in
the first matrix is the same as the number of rows in the second matrix. The result is a matrix with the
same number of rows as the first matrix and the same number of columns as the second matrix. Thus
[ C ] PxS =[ A ] PxQ [ B ] QxS
is possible while
[ A ] PxQ [ B ] PxQ
is not. Multiplication of two matrices is defined as
å
=
k Q
c i , j k 1 ( ai , k )( bk , j )
for each element in the resulting matrix. Note that, in general, matrix multiplication is not
commutative.
(8) Matrix algebra in MATLAB.
- Confusing matrix algebra operators with element-by-element operators can result in big mistakes.
Define (enter) the following matrices at the MATLAB command prompt.
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Amat
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Bmat
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ê
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Cmat
Try performing the following mathematical operations
Amat + Bmat Amat + Cmat Bmat − Amat Cmat − Bmat
Amat * Bmat Amat * Cmat Bmat * Amat Cmat * Amat
What are the results? What error messages are generated?
Exercises : Perform the following operations using array arithmetic where appropriate.
1. Create the following three matrices:
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A
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B
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C
a. Show that matrix addition is commutative by computing A+B and B+A.
b. Show that matrix addition is associative by computing A+(B+C) and (A+B)+C.
c. Show that scalar-matrix multiplication is distributive by computing 5(A+C) and 5A+5C.
d. Show that matrix multiplication is distributive by computing A(B+C) and AB+B*C.
2. Use the matrices from exercise 1 to answer the following
a. Does AB = BA?
b. Does (AB)C = A(BC)?
c. Does (A*B)t^ = At^ *Bt^ (t means transpose)?
d. Does (A+B)
t
= A
t
+B
t
e. Does AB = A.B?
Recap : You should have learned
- How to declare row and column vectors in MATLAB
- How to declare a vector with evenly spaced elements (two methods)
