Finding a Unit Vector v for a Matrix A such that the 2-norm of AV is equal to the 2-norm of A

Question

I have been working on the following problem:

Let A be the following 2x2 matrix: A = [1 1; 0 1] (MATLAB notation)

Find the 2-norm of A and a unit vector v such that the 2-norm of Av = the 2-norm of A.

I know that the 2-norm of A is (1+sqrt(5))/2, but I can't for the life of me figure out how to find such a vector v. How should I go about this?

Answer 1

The 2-norm of a matrix is just its largest singular value. So, your $v$ is the associated singular vector.

A=[1 1;0 1]; [U,S,V]=svd(A); v=V(:,1), [norm(A*v), S(1), norm(A)]

v =

0.5257

0.8507

ans =

1.6180 1.6180 1.6180