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More About Matrices and Arrays

Posted by SUYOG PATIL on 8:29 PM in , , ,
Generating Matrices
MATLAB software provides four functions that generate basic matrices.
zeros All zeros
ones All ones
rand Uniformly distributed random elements
randn Normally distributed random elements
Here are some examples:
Z = zeros(2,4)
Z =
0 0 0 0
0 0 0 0
F = 5*ones(3,3)
F =
5 5 5
5 5 5
5 5 5
N = fix(10*rand(1,10))
N =
9 2 6 4 8 7 4 0 8 4
R = randn(4,4)
R =
0.6353 0.0860 -0.3210 -1.2316
-0.6014 -2.0046 1.2366 1.0556
0.5512 -0.4931 -0.6313 -0.1132
-1.0998 0.4620 -2.3252 0.3792
The load Function
The load function reads binary files containing matrices generated by earlier
MATLAB sessions, or reads text files containing numeric data. The text file
should be organized as a rectangular table of numbers, separated by blanks,
with one row per line, and an equal number of elements in each row. For
example, outside of MATLAB, create a text file containing these four lines:
16.0 3.0 2.0 13.0
5.0 10.0 11.0 8.0
9.0 6.0 7.0 12.0
4.0 15.0 14.0 1.0
Save the file as magik.dat in the current directory. The statement
load magik.dat
reads the file and creates a variable, magik, containing the example matrix.
An easy way to read data into MATLAB from many text or binary formats
is to use the Import Wizard.
Saving Code to a File
You can create matrices using text files containing MATLAB code. Use the
MATLAB Editor or another text editor to create a file containing the same
statements you would type at the MATLAB command line. Save the file
under a name that ends in .m.
For example, create a file in the current directory named magik.m containing
these five lines:

A = [16.0 3.0 2.0 13.0
5.0 10.0 11.0 8.0
9.0 6.0 7.0 12.0
4.0 15.0 14.0 1.0 ];
The statement
magik
reads the file and creates a variable, A, containing the example matrix.
Concatenation
Concatenation is the process of joining small matrices to make bigger ones. In
fact, you made your first matrix by concatenating its individual elements. The
pair of square brackets, [], is the concatenation operator. For an example,
start with the 4-by-4 magic square, A, and form
B = [A A+32; A+48 A+16]
The result is an 8-by-8 matrix, obtained by joining the four submatrices:
B =
16 3 2 13 48 35 34 45
5 10 11 8 37 42 43 40
9 6 7 12 41 38 39 44
4 15 14 1 36 47 46 33
64 51 50 61 32 19 18 29
53 58 59 56 21 26 27 24
57 54 55 60 25 22 23 28
52 63 62 49 20 31 30 17
This matrix is halfway to being another magic square. Its elements are a
rearrangement of the integers 1:64. Its column sums are the correct value
for an 8-by-8 magic square:
sum(B)
ans =
260 260 260 260 260 260 260 260
But its row sums, sum(B')', are not all the same. Further manipulation is
necessary to make this a valid 8-by-8 magic square.
Deleting Rows and Columns
You can delete rows and columns from a matrix using just a pair of square
brackets. Start with
X = A;
Then, to delete the second column of X, use
X(:,2) = []
This changes X to
X =
16 2 13
5 11 8
9 7 12
4 14 1
If you delete a single element from a matrix, the result is not a matrix
anymore. So, expressions like
X(1,2) = []
result in an error. However, using a single subscript deletes a single element,
or sequence of elements, and reshapes the remaining elements into a row
vector. So
X(2:2:10) = []
results in
X =
16 9 2 7 13 12 1
Linear Algebra
Informally, the terms matrix and array are often used interchangeably. More
precisely, a matrix is a two-dimensional numeric array that represents a
linear transformation. The mathematical operations defined on matrices are
the subject of linear algebra.
Dürer’s magic square
A = [16 3 2 13
5 10 11 8
9 6 7 12
4 15 14 1]
provides several examples that give a taste of MATLAB matrix operations.
You have already seen the matrix transpose, A'. Adding a matrix to its
transpose produces a symmetric matrix:
A + A'
ans =
32 8 11 17
8 20 17 23
11 17 14 26
17 23 26 2
The multiplication symbol, *, denotes the matrix multiplication involving
inner products between rows and columns. Multiplying the transpose of a
matrix by the original matrix also produces a symmetric matrix:
A'*A
ans =
378 212 206 360
212 370 368 206
206 368 370 212
360 206 212 378
The determinant of this particular matrix happens to be zero, indicating
that the matrix is singular:
d = det(A)
d =
0
The reduced row echelon form of A is not the identity:
R = rref(A)
R =
1 0 0 1
0 1 0 -3
0 0 1 3
0 0 0 0
Because the matrix is singular, it does not have an inverse. If you try to
compute the inverse with
X = inv(A)
you will get a warning message:
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 9.796086e-018.
Roundoff error has prevented the matrix inversion algorithm from detecting
exact singularity. But the value of rcond, which stands for reciprocal
condition estimate, is on the order of eps, the floating-point relative precision,
so the computed inverse is unlikely to be of much use.
The eigenvalues of the magic square are interesting:
e = eig(A)
e =
34.0000
8.0000
0.0000
-8.0000
One of the eigenvalues is zero, which is another consequence of singularity.
The largest eigenvalue is 34, the magic sum. That sum results because the
vector of all ones is an eigenvector:
v = ones(4,1)
v =
1
1
1
1
A*v
ans =
34
34
34
34
When a magic square is scaled by its magic sum,
P = A/34
the result is a doubly stochastic matrix whose row and column sums are all 1:
P =
0.4706 0.0882 0.0588 0.3824
0.1471 0.2941 0.3235 0.2353
0.2647 0.1765 0.2059 0.3529
0.1176 0.4412 0.4118 0.0294
Such matrices represent the transition probabilities in a Markov process.
Repeated powers of the matrix represent repeated steps of the process. For
this example, the fifth power
P^5
is
0.2507 0.2495 0.2494 0.2504
0.2497 0.2501 0.2502 0.2500
0.2500 0.2498 0.2499 0.2503
0.2496 0.2506 0.2505 0.2493
This shows that as k approaches infinity, all the elements in the k th power,p^k approach 1/4
Finally, the coefficients in the characteristic polynomial
poly(A)
are
1 -34 -64 2176 0
These coefficients indicate that the characteristic polynomial is 
The constant term is zero because the matrix is singular. The coefficient of
the cubic term is -34 because the matrix is magic!

Arrays
When they are taken away from the world of linear algebra, matrices become
two-dimensional numeric arrays. Arithmetic operations on arrays are
done element by element. This means that addition and subtraction are
the same for arrays and matrices, but that multiplicative operations are
different. MATLAB uses a dot, or decimal point, as part of the notation for
multiplicative array operations.
The list of operators includes
+ Addition
- Subtraction
.* Element-by-element multiplication
./ Element-by-element division
.\ Element-by-element left division
.^ Element-by-element power
.' Unconjugated array transpose
If the Dürer magic square is multiplied by itself with array multiplication
A.*A
the result is an array containing the squares of the integers from 1 to 16,
in an unusual order:
ans =
256 9 4 169
25 100 121 64
81 36 49 144
16 225 196 1
Building Tables
Array operations are useful for building tables. Suppose n is the column vector
n = (0:9)';
Then
pows = [n n.^2 2.^n]
builds a table of squares and powers of 2:
pows =
0 0 1
1 1 2
2 4 4
3 9 8
4 16 16
5 25 32
6 36 64
7 49 128
8 64 256
9 81 512
The elementary math functions operate on arrays element by element. So
format short g
x = (1:0.1:2)';
logs = [x log10(x)]
builds a table of logarithms.
logs =
1.0 0
1.1 0.04139
1.2 0.07918
1.3 0.11394
1.4 0.14613
1.5 0.17609
1.6 0.20412
1.7 0.23045
1.8 0.25527
1.9 0.27875
2.0 0.30103

Multivariate Data
MATLAB uses column-oriented analysis for multivariate statistical data.
Each column in a data set represents a variable and each row an observation.
The (i,j)th element is the ith observation of the jth variable.
As an example, consider a data set with three variables:
• Heart rate
• Weight
• Hours of exercise per week
For five observations, the resulting array might look like
D = [ 72 134 3.2
81 201 3.5
69 156 7.1
82 148 2.4
75 170 1.2 ]
The first row contains the heart rate, weight, and exercise hours for patient 1,
the second row contains the data for patient 2, and so on. Now you can apply
many MATLAB data analysis functions to this data set. For example, to
obtain the mean and standard deviation of each column, use
mu = mean(D), sigma = std(D)
mu =
75.8 161.8 3.48
sigma =
5.6303 25.499 2.2107
For a list of the data analysis functions available in MATLAB, type
help datafun
If you have access to the Statistics Toolbox™ software, type
help stats


Scalar Expansion
Matrices and scalars can be combined in several different ways. For example,
a scalar is subtracted from a matrix by subtracting it from each element. The
average value of the elements in our magic square is 8.5, so
B = A - 8.5
forms a matrix whose column sums are zero:
B =
7.5 -5.5 -6.5 4.5
-3.5 1.5 2.5 -0.5
0.5 -2.5 -1.5 3.5
-4.5 6.5 5.5 -7.5
sum(B)
ans =
0 0 0 0
With scalar expansion, MATLAB assigns a specified scalar to all indices in a
range. For example,
B(1:2,2:3) = 0
zeroes out a portion of B:
B =
7.5 0 0 4.5
-3.5 0 0 -0.5
0.5 -2.5 -1.5 3.5
-4.5 6.5 5.5 -7.5



Logical Subscripting
The logical vectors created from logical and relational operations can be used
to reference subarrays. Suppose X is an ordinary matrix and L is a matrix of
the same size that is the result of some logical operation. Then X(L) specifies
the elements of X where the elements of L are nonzero.This kind of subscripting can be done in one step by specifying the logical
operation as the subscripting expression. Suppose you have the following
set of data:
x = [2.1 1.7 1.6 1.5 NaN 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8];
The NaN is a marker for a missing observation, such as a failure to respond to
an item on a questionnaire. To remove the missing data with logical indexing,
use isfinite(x), which is true for all finite numerical values and false for
NaN and Inf:
x = x(isfinite(x))
x =
2.1 1.7 1.6 1.5 1.9 1.8 1.5 5.1 1.8 1.4 2.2 1.6 1.8
Now there is one observation, 5.1, which seems to be very different from the
others. It is an outlier. The following statement removes outliers, in this case
those elements more than three standard deviations from the mean:
x = x(abs(x-mean(x)) <= 3*std(x))
x =
2.1 1.7 1.6 1.5 1.9 1.8 1.5 1.8 1.4 2.2 1.6 1.8
For another example, highlight the location of the prime numbers in Dürer’s
magic square by using logical indexing and scalar expansion to set the
nonprimes to 0.
A(~isprime(A)) = 0
A =
0 3 2 13
5 0 11 0
0 0 7 0
0 0 0 0
The find Function
The find function determines the indices of array elements that meet a given
logical condition. In its simplest form, find returns a column vector of indices.
Transpose that vector to obtain a row vector of indices. For example, start with Dürer’s magic square.
k = find(isprime(A))'
picks out the locations, using one-dimensional indexing, of the primes in the
magic square:
k =
2 5 9 10 11 13
Display those primes, as a row vector in the order determined by k, with
A(k)
ans =
5 3 2 11 7 13
When you use k as a left-hand-side index in an assignment statement, the
matrix structure is preserved:
A(k) = NaN
A =
16      NaN    NaN    NaN
NaN  10        NaN    8
9        6          NaN    12
4       15         14        1



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