A = [ 2 0; 1 -1 ]
A =
2 0 1 -1
The default norm returned by the norm command is the 2-norm.
twonorm = norm(A)
twonorm =
2.2882e+00
You can get the 1-norm as well.
onenorm = norm(A,1)
onenorm =
3
The 1-norm is equivalent to
max( sum(abs(A),1) ) % sum along rows (1st matrix dimension)
ans =
3
Similarly, we can get the ∞-norm and check our formula for it.
infnorm = norm(A,inf)
infnorm =
2
max( sum(abs(A),2) ) % sum along columns (2nd matrix dimension)
ans =
2
Here we illustrate the geometric interpretation of the 2-norm. First, we will sample a lot of vectors on the unit circle in .
theta = linspace(0,2*pi,601);
x = [ cos(theta); sin(theta) ]; % 601 unit columns
subplot(1,2,1), plot(x(1,:),x(2,:)), axis equal
title('Unit circle in 2-norm')
xlabel('x_1'), ylabel('x_2')
We can apply to every column of simply by using
Ax = A*x;
We superimpose the image of the unit circle with the circle whose radius is , and display multiple plots with the subplot command.
subplot(1,2,2), plot(Ax(1,:),Ax(2,:)), axis equal
hold on, plot(twonorm*x(1,:),twonorm*x(2,:),'--')
title('Image of Ax, with ||A||')
xlabel('x_1'), ylabel('x_2')