initial commit / add all books

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+disp('To find a basis for the eigenspace')

+disp('Matrix A=')

+a=[4 0 1;-2 1 0;-2 0 1]

+disp(a)

+disp('for lambda=1')

+disp('A-1I=')

+b=a-eye(3,3)

+disp(b)

+disp('solving (A-I)x=0, we get')

+disp('-2*x1=0 and 3*x1+x3=0')

+disp('therefore, x1=x3=0')

+disp('which leaves x2 as a free variable')

+disp('Hence a basis for the eigen space is:')

+disp([0;1;0])

+disp('for lambda=2')

+disp('A-2I=')

+b=a-2*eye(3,3)

+disp(b)

+disp('performing row operations on the augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+c(2,:)=c(2,:)+c(1,:)

+c(3,:)=c(3,:)+c(1,:)

+disp(c)

+c(1,:)=c(1,:)/c(2,2)

+disp(c)

+disp('We can see that x3 is a free variable')

+disp('x2=x3 and x1=-.05*x3')

+disp('Hence, a basis for the eigenspace is:')

+disp([-.5;1;1])

+disp('for lambda=3')

+disp('A-3I=')

+b=a-3*eye(3,3)

+disp(b)

+disp('performing row operations on the augmented matrix')

+c=[b [0;0;0]]

+disp(c)

+c(2,:)=c(2,:)+2*c(1,:)

+c(3,:)=c(3,:)+2*c(1,:)

+disp(c)

+c(2,:)=c(2,:)/2

+disp(c)

+disp('Again x3 is a free variable')

+disp('x1=-x3 and x2=x3')

+disp('Hence, a basis for the eigenspace is:')

+disp([-1;1;1])
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