This matrix describes the votes on bills in the 111th session of the United States Senate. (The data set was obtained from voteview.com.) Each row is one senator and each column is a vote item.
load voting % get from the book's website
[m,n] = size(A);
If we visualize the votes (white is ``yea,'' black is ``nay,'' and gray is anything else), we can see great similarity between many rows, reflecting party unity.
imagesc(A)
colormap gray
title('Votes in 111th U.S. Senate') % ignore this line
ylabel('senator'), xlabel('bill') % ignore this line
We use singular value ``energy'' to quantify the decay rate of the values.
[U,S,V] = svd(A);
sigma = diag(S);
tau = cumsum(sigma.^2) / sum(sigma.^2);
plot(tau(1:16),'.')
xlabel('k'), ylabel('\tau_k') % ignore this line
title('Fraction of singular value energy') % ignore this line
The first and second singular triples contain about 58% and 17% respectively of the energy of the matrix. All others have far less effect, suggesting that the information is primarily two-dimensional. The first left and right singular vectors also contain interesting structure.
subplot(211), plot(U(:,1),'.')
axis tight % ignore this line
xlabel('senator number'), title('left singular vector') % ignore this line
subplot(212), plot(V(:,1),'.')
axis tight % ignore this line
xlabel('bill number'), title('right singular vector') % ignore this line
Both vectors have values greatly clustered near 
for a constant C. These can be roughly interpreted as how partisan a particular senator or bill was, and for which political party.
for a constant C. These can be roughly interpreted as how partisan a particular senator or bill was, and for which political party. Projecting the senators' vectors into the first two 
-coordinates gives a particularly nice way to reduce them to two dimensions. Political scientists label these dimensions ``partisanship'' and ``bipartisanship.'' Here we color them by actual party affiliation (also given in the data file): red for Republican, blue for Democrat, and black for independent.
-coordinates gives a particularly nice way to reduce them to two dimensions. Political scientists label these dimensions ``partisanship'' and ``bipartisanship.'' Here we color them by actual party affiliation (also given in the data file): red for Republican, blue for Democrat, and black for independent. clf
x1 = V(:,1)'*A'; x2 = V(:,2)'*A';
scatter(x1(Dem),x2(Dem),20,'b'), hold on
scatter(x1(Rep),x2(Rep),20,'r')
scatter(x1(Ind),x2(Ind),20,'k')
xlabel('partisanship'), ylabel('bipartisanship') % ignore this line
title('111th US Senate in 2D')
