In linear algebra, a Hilbert matrix,
introduced by Hilbert (1894),
is a square
matrix with entries being the unit fractions
For example, this is the 10 × 10 Hilbert matrix:
hilb(10); %
Hibert Matrix
h = hilb(10);
plot(h)
title('Hilbert
Matix')
axis off
hi = invhilb(10)
% Inverse Hilbert matrix
hi =
1.0e+12 *
0.0000
-0.0000 0.0000 -0.0000
0.0000 -0.0000 0.0000
-0.0000 0.0000 -0.0000
-0.0000
0.0000 -0.0000 0.0000
-0.0002 0.0005 -0.0008
0.0008 -0.0004 0.0001
0.0000
-0.0000 0.0001 -0.0010
0.0043 -0.0112 0.0178
-0.0166 0.0085 -0.0018
-0.0000
0.0000 -0.0010 0.0082
-0.0379 0.1010 -0.1616
0.1529 -0.0788 0.0171
0.0000
-0.0002 0.0043 -0.0379
0.1768 -0.4772 0.7713
-0.7359 0.3821 -0.0832
-0.0000
0.0005 -0.0112 0.1010
-0.4772 1.3015 -2.1210
2.0378 -1.0644 0.2330
0.0000
-0.0008 0.0178 -0.1616
0.7713 -2.1210 3.4807
-3.3640 1.7661 -0.3884
-0.0000
0.0008 -0.0166 0.1529
-0.7359 2.0378 -3.3640
3.2679 -1.7233 0.3804
0.0000
-0.0004 0.0085 -0.0788
0.3821 -1.0644 1.7661
-1.7233 0.9123 -0.2021
-0.0000
0.0001 -0.0018 0.0171
-0.0832 0.2330 -0.3884
0.3804 -0.2021 0.0449
plot(hi)
title('Inverse
Hilbert Matix')
axis off
%In recreational
mathematics, a magic
square is an arrangement of
numbers (usually integers) in %a square grid, where the numbers in each row, and in each column, and
the numbers in the forward %and backward main diagonals, all add up to the same
number. A magic square has the same number %of rows as it has columns, and in
conventional math notation, "n" stands for the number of rows (and
%columns) it has. Thus, a magic square always contains n2 numbers, and its
size (the number of rows %[and columns] it has) is described as being "of
order n". A magic square
that contains the integers %from 1 to n2 is called a normal magic square.
m = magic(10)%
Magic Matrix
m =
92
99 1 8
15 67 74
51 58 40
98
80 7 14
16 73 55
57 64 41
4
81 88 20
22 54 56
63 70 47
85
87 19 21
3 60 62
69 71 28
86
93 25 2
9 61 68
75 52 34
17
24 76 83
90 42 49
26 33 65
23
5 82 89
91 48 30
32 39 66
79
6 13 95
97 29 31
38 45 72
10
12 94 96
78 35 37
44 46 53
11
18 100 77
84 36 43
50 27 59
plot(m)
plot(m)
title('Magic
Matrix')
axis off
In mathematics, particularly matrix
theory and combinatory, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are three ways to achieve this: as
either an upper-triangular matrix, a
lower-triangular matrix, or asymmetric matrix
pascal(10) %
Pascal Matrix
ans =
1 1 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
1 3 6 10 15 21 28
36 45 55
1 4 10 20 35 56 84 120 165 220
1 5 15 35 70 126 210 330
495 715
1 6 21 56 126 252 462 792 1287 2002
1 7 28 84 210 462 924 1716 3003 5005
1 8 36 120 330 792 1716 3432 6435 11440
1 9 45 165 495 1287 3003 6435 12870 24310
1
10 55 220 715 2002 5005 11440 24310 48620
plot(ans)
title('Pascal
Matrix')
axis off
%A matrix equation of the form
%
%is
called a Toeplitz system if A is a Toeplitz matrix. If A is an
Toeplitz
matrix, then the %system has only 2n−1 degrees
of freedom, rather than n2.
We might therefore expect that the %solution of a Toeplitz system would be
easier, and indeed that is the case.
t = toeplitz(10)
% Toeplitz Matrix
t =
10
A Vandermonde matrix is a type of matrix that
arises in the polynomial
least squares fitting, Lagrange interpolating
polynomials (Hoffman and Kunze p. 114), and the reconstruction of
a statistical
distribution from the distribution's moments
v = vander(10) %
Vandermonde Matrix
v =
1
In linear
algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues. It
is named after the British mathematician James
H. Wilkinson
w = wilkinson(10)
% Wilkinsons eigen value test matrix
w =
4.5000
1.0000 0 0 0 0 0 0 0 0
1.0000
3.5000 1.0000 0 0 0 0 0
0 0
0
1.0000 2.5000 1.0000 0 0 0 0 0 0
0 0
1.0000 1.5000 1.0000 0 0 0 0 0
0 0 0
1.0000 0.5000 1.0000 0 0 0 0
0 0 0 0
1.0000 0.5000 1.0000 0 0 0
0 0 0 0 0
1.0000 1.5000 1.0000 0 0
0 0 0 0 0 0
1.0000 2.5000 1.0000 0
0 0 0 0 0 0 0
1.0000 3.5000 1.0000
0 0 0 0
0 0 0 0
1.0000 4.5000
plot(w)
title('Wilkinson
Matrix')
axis off
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