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+disp('PD decomposition of a matrix A')

+a=[3 -2 4;-2 6 2;4 2 3]

+disp(a,'A=')

+disp('Eigenvalues of A are')

+eig=spec(a)

+disp(eig)

+disp(eig(2,1),'for lambda =')

+disp('A-(lambda)I=')

+b=a-eig(2,1)*eye(3,3)

+disp(b)

+disp('To find eigenvector, form an augmented matrix')

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

+disp(c)

+disp('performing row operations')

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

+c(3,:)=c(3,:)-(c(3,1)/c(1,1))*c(1,:)

+disp(c)

+disp('With x2 and x3 as free variables, we get two vectors.')

+disp('x1=-.5x2+x3')

+disp('Thus, the two vectors are')

+v1=[-1;2;0]

+v2=[1;0;1]

+disp(v2,v1)

+disp('Orthogonalizing v1 and v2')

+disp('Let x1=v1')

+disp('x2=v2-((v2.v1)/(v1.v1))*v1')

+x1=v1

+a1=v2'*v1

+a2=v1'*v1

+x2=v2-(a1/a2)*v1

+x1=x1/(sqrt(x1(1,1)^2+x1(2,1)^2+x1(3,1)^2))

+x1=x2/(sqrt(x2(1,1)^2+x2(2,1)^2+x2(3,1)^2))

+disp('An orthonormal basis is:')

+disp(x2,x1)

+disp(eig(1,1),'for lambda=')

+disp('A-(lambda)I=')

+b=a-eig(1,1)*eye(3,3)

+disp(b)

+disp('To find eigenvector, form an augmented matrix')

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

+disp(c)

+disp('performing row operations')

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

+c(3,:)=c(3,:)-(c(3,1)/c(1,1))*c(1,:)

+disp(c)

+c(3,:)=c(3,:)-(c(3,2)/c(2,2))*c(2,:)

+disp(c)

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

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

+disp(c)

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

+disp(c)

+disp('With x3 as free variable')

+disp('x1=x3 and x2=-.5x3')

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

+v3=[1;-.5;1]

+disp(v3)

+disp('upon normalizing')

+v3=v3/(sqrt(v3(1,1)^2+v3(2,1)^2+v3(3,1)^2))

+disp(v3)

+disp('Thus, matrix P=')

+disp([x1 x2 v3])

+disp('Corresponding matrix D=')

+disp([eig(2,1) 0 0;0 eig(3,1) 0;0 0 eig(1,1)])
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