Question

Do the following using Matlab: Let A be a matrix where each row corresponds to an article, and each column to how often a specific word appears in the article. Patterns among the articles and words can often be identified by analyzing the singular values and vectors of the singular value decomposition of A.

(a) Use the Matlab svd function to find the singular values of the following . Note this problem does not ask you to print out the entire single value decomposition, only the singular values.

matriar

A = [9 0 3 4 0 0 0 0 0 4 1 0 0 1 3 2 1 8 0 1

8 0 3 5 0 0 1 0 0 3 1 0 0 2 3 2 1 7 0 1

0 1 0 1 2 3 4 5 3 1 0 0 0 0 0 1 1 0 2 7

7 0 3 4 0 0 1 0 0 5 1 0 0 0 2 2 1 8 1 1

0 1 2 0 2 3 4 5 2 1 0 1 0 0 0 1 1 0 2 7

9 0 4 5 0 0 1 0 0 4 1 0 1 0 2 2 0 7 0 1

0 1 0 2 4 3 5 6 3 2 0 0 0 0 0 1 0 0 2 6]

(b) Have Matlab print the ratio of the largest singular value to the smallest singular value. Also have Matlab print out the condition number of A (recall the Matlab cond(A) function returns the condition number of A).

(c) Use the Matlab svds functions with arguments A and 2 to find the singular value decomposition associated with the largest two singular values of A. For this part have Matlab return U, ?, and V matrices. (Here, U, ?, and V refer to the portion of those matrices returned by svds when the second argument is 2.)

(d) Have Matlab multiply $U\Sigma V^{T}$ from part (c), print the product, and then compare the result to the original A matrix by printing the norm of $U\Sigma V^{T}-A$ . In this problem, make sure you use the U, ?, and V from part (c).

For this problem, turn in a diary or script — showing both the Matlab statements used and Matlab’s output — for parts (a) through (d)).

Answer 1

A = [9 0 3 4 0 0 0 0 0 4 1 0 0 1 3 2 1 8 0 1;
8 0 3 5 0 0 1 0 0 3 1 0 0 2 3 2 1 7 0 1;
0 1 0 1 2 3 4 5 3 1 0 0 0 0 0 1 1 0 2 7;
7 0 3 4 0 0 1 0 0 5 1 0 0 0 2 2 1 8 1 1;
0 1 2 0 2 3 4 5 2 1 0 1 0 0 0 1 1 0 2 7;
9 0 4 5 0 0 1 0 0 4 1 0 1 0 2 2 0 7 0 1;
0 1 0 2 4 3 5 6 3 2 0 0 0 0 0 1 0 0 2 6];

%% part a
s = svd(A); % print singular values of matrix A
display(s);
% output
% s =
%
% 27.5875
% 18.8616
% 3.0180
% 2.5743
% 2.2455
% 1.4063
% 1.1891

%% part b
r = s(1)/s(end);
display(r); % print the ratio of largest to smallest singular value
% output
% r =
%
% 23.2005
%% part c
[U,S,V] = svd(A);
% extracting 2 colm from U
u2 = U(1:end,1:2);
% extracting 2 row and 2 colm from S
s2 = S(1:2,1:2);
% extracting 2 colms from V
v2 = V(1:end,1:2);
% ans = u2, s2, v2
display(u2);
display(s2);
display(v2);
% output
% u2 =
%
% -0.5056 -0.1387
% -0.4723 -0.1026
% -0.1162 0.5537
% -0.4715 -0.0835
% -0.1203 0.5413
% -0.5013 -0.1096
% -0.1406 0.5931
% s2 =
%
% 27.5875 0
% 0 18.8616
% v2 =
%
% -0.5851 -0.1929
% -0.0137 0.0895
% -0.2390 -0.0175
% -0.3325 -0.0111
% -0.0375 0.2419
% -0.0410 0.2685
% -0.1121 0.3738
% -0.0734 0.4789
% -0.0366 0.2398
% -0.3016 0.0299
% -0.0707 -0.0230
% -0.0044 0.0287
% -0.0182 -0.0058
% -0.0526 -0.0182
% -0.1769 -0.0588
% -0.1551 0.0434
% -0.0611 0.0408
% -0.5304 -0.1729
% -0.0444 0.1746
% -0.1613 0.5720
%% part d
m = u2*s2*v2'; % calculate U*S*V' for 2 singular value
d = norm(m - A); % calculate norm of diff os U*S*V' and A
display(d); % finally print the norm
% output
% d =
%
% 3.0180