CHAPTER 2: MATRIX COMPUTATIONS AND LINEAR ALGEBRA
Lecture 2.1: Two-dimensional
arrays (matrices) and matrix operations
In linear algebra, a matrix is a rectangular array
of elements (numbers). If a matrix A = [ ai,j ] has m
rows and n columns, it has m-by-n elements ai,j where the first index i runs for rows of A and
the second index j runs for columns of A. MATLAB
matrices are direct and convenient numerical realization of matrices in linear
algebra.
A = [ 0.1 0.2 0.3 0.4 ;
-0.1,-0.2,-0.3,-0.4 ; 0 1 0 1 ]
% Notice that rows in
MATLAB matrices are separated by the sign ";"
% Notice that elements at each row can be separated by spaces and commas
0.1000 0.2000
0.3000 0.4000
-0.1000 -0.2000
-0.3000 -0.4000
0 1.0000 0 1.0000
A1 = [ 0.1 0.2 0.3 0.4 ; 0 1 0 1 0] ;
% Number of elements in each row must be identical
All rows in the bracketed expression must have the same
number of columns.
The colon sign ":" represents all
elements of a row or a column of a MATLAB matrix.
A(:,1), A(:,4) % columns of A are column-vectors
0.1000
-0.1000
0
ans =
0.4000
-0.4000
1.0000
A(1,:), A(3,:) % rows of A are row-vectors
0.1000 0.2000
0.3000 0.4000
ans =
0 1 0 1
A(1,2), A(3,3) % individual elements of A are scalar variables
0.2000
ans =
0
Matrix operations:
·
all vector operations, such as pointwise additions, subtractions,
multiplications, divisions, and powers of matrices of the same size
A = [ 1 2 3 4 ; -1,-2,-3,-4 ];
B = [ 4 3 2 1 ; -4,-3,-2,-1
];
C1 = A + B, C2 = A.*B, C3 = A./B, C4 = A.^B
% Pointwise matrix operations are preferable compared to double
loops in i,j
% because of (i) clear mathematical notations, (ii) computational
efficiency
5 5
5 5
-5 -5
-5 -5
C2 =
4 6
6 4
4 6
6 4
C3 =
0.2500 0.6667 1.5000 4.0000
0.2500 0.6667
1.5000 4.0000
C4 =
1.0000 8.0000
9.0000 4.0000
1.0000 -0.1250 0.1111 -0.2500
·
size:
outputs numbers of rows and columns
nCol = 4
·
transpose: A', outputs a transposed matrix AT
2 -2
3 -3
4 -4
ans =
·
elementary matrices:
- zeros: outputs a null matrix
with all elements being zeros
zeros(2,6) % a null 2-by-6 matrix
zeros(3) % a null 3-by-3 matrix
0 0 0
0 0 0
ans = 0
0 0
0
0 0
0 0 0
- ones: outputs an matrix with
all elements being ones
ones(2,6) % a 2-by-6 matrix of ones
ones(3) % a 3-by-3 matrix
of ones
1 1 1
1 1 1
1 1 1
1 1 1
ans =
1 1 1
1 1 1
1 1 1
- eye: outputs an identity
matrix (a diagonal matrix)
1 0 0
0 0 0
0 1 0
0 0 0
ans =
1 0 0
0 1 0
0 0 1
- rand, randn: outputs a random matrix
with all elements being random numbers
rand(2,6) % random
numbers are uniformly distributed in the interval [0,1]
randn(3) % random numbers are normally distributed with mean zero and variance one
0.9501 0.6068 0.8913
0.4565 0.8214 0.6154
0.2311 0.4860 0.7621
0.0185 0.4447 0.7919
ans =
-0.4326 0.2877 1.1892
-1.6656 -1.1465 -0.0376
0.1253 1.1909 0.3273
·
special matrices:
- hilb: outputs the Hilbert
matrix with elements ai,j = 1/(i+j-1)
C1 = hilb(6) % Hilbert matrices are examples of ill-conditioned matrices
C1 = 1.0000 0.5000
0.3333 0.2500 0.2000
0.5000 0.3333
0.2500 0.2000 0.1667
0.3333 0.2500 0.2000
0.1667 0.1429
0.2500 0.2000
0.1667 0.1429 0.1250
0.2000 0.1667 0.1429 0.1250 0.1111
- pascal: outputs the Pascal
matrix with integer entries, made up from Pascal's triangle: ai,j
= ai-1,j + ai,j-1
% Pascal matrices produce a
table of binomial coefficients
% Coefficients at the k-th
anti-diagonal of Pascal matrices produce
% coefficients of the binomial expansion: (x + y)^(k-1)
1 2
3 4 5 6
1 3
6 10 15 21
1 4
10 20 35 56
1 5
15 35 70
126
1 6 21 56 126 252
- vander: outputs the Vandermonde
matrix whose columns are powers of a vector, i.e. ai,j =
vin-j
32
16 8 4 2 1
243
81 27 9 3 1
1024
256 64 16 4 1
3125 625 125
25 5 1
7776 1296 216 36 6 1
- toeplitz: outputs the Toeplitz
matrices
r = [ 1 2 3 4 5 6 ]; s = [ -1,-2,-3,-4, -5];
C4 = toeplitz(r,s) % r and s are the column and
row vectors of C4
C5 = toeplitz(r) % the symmetric Toeplitz matrix
rInv = r(length(r):-1:2); rInv =
[r(length(r)),rInv];
C6 = toeplitz(r,rInv)' % the circulant matrix
Warning: Column wins diagonal conflict.
> In
c:\progra~1\matlab6\toolbox\matlab\elmat\toeplitz.m at line 18
C4 =
1 -2 -3
-4 -5
2 1 -2
-3 -4
3 2 1
-2 -3
4 3 2
1 -2
5 4 3
2 1
6 5 4
3 2
C5 =
1 2 3 4 5 6
2 1 2
3 4 5
3 2 1
2 3 4
4 3 2
1 2 3
5 4 3
2 1 2
6 5 4
3 2 1
Warning: Column wins
diagonal conflict.
> In c:\progra~1\matlab6\toolbox\matlab\elmat\toeplitz.m
at line 18
C6 =
1 2 3
4 5 6
6 1 2
3 4 5
5 6 1
2 3 4
4 5 6
1 2 3
3 4 5
6 1 2
2 3 4
5 6 1
- hadamard, hankel: more
specialized matrices
·
matrix-vector multiplications
A = [ 1 2 3 1 ; 0 1 4 2 ; 3 0 2 3 ], x = [ 1; 2; 1; 1; ]
0 1 4
2
3 0 2
3
x = 1
2
1
1
% the number of columns
of A must be equal to the number of rows of x
y1 = A*x
% MATLAB matrix multiplication operator
% a row-oriented loop:
[n,m] = size(A);
for k = 1:n, y2(k) = A(k,:)*x; end
y2
% a column-oriented loop
y3 = zeros(n,1);
for j = 1 : m, y3 = y3 + A(:,j)*x(j); end
y3
8
8
y2 = 9
8 8
y3 = 9
8
8
·
matrix-matrix mutiplications:
If A is a n-by-m
matrix, B is a m-by-q matrix, then
the matrix multiplication makes sense and C = A*B is a n-by-q
matrix with elements: ci,j = 
ai,k bk,j . The other product B*A
is defined if n = q. For a square matrices of the same size, both
products are defined but they are not equal in a general case: A*B 
B*A.
A = [ -2 1 0 ; 1,-2,1 ; 0,1,-2];
B = [1 0 1 ; 0 1 0; 1 0 1];
C1 = A*B, C2 = B*A
-2 1 -2
2 -2 2
-2 1 -2
C2 =
-2 2 -2
1 -2 1
-2 2 -2
n = 3;
for k = 1 : n
for j = 1
: n
C3(k,j)
= A(k,:)*B(:,k); % the dot product algorithm
end
end
C3
C4 = zeros(n,n);
for j = 1 : n
for p = 1
: n
C4(:,j)
= C4(:,j) + A(:,p)*B(k,j); % the column-oriented algorithm
end
end
C4
C5 = zeros(n,n);
for p = 1 : n
C5 = C5 +
A(:,k)*B(k,:); % the outer-product oriented algorithm
end
C5
-2 -2 -2
-2 -2 -2
-2 -2 -2
C4 =
-1 0 -1
0 0 0
-1 0 -1
C5 =
0 0 0
3 0 3
-6 0 -6