Eigenvalues in octave with eig()

Question

Consider the real, symmetric matrix

S = (2, 1; 1, 2)

From the characteristic equation |S - λ I|, we have the quadratic (2-λ)^2 - 1 = 0, whose solutions yield the eigenvalues 3 and 1. The corresponding eigenvectors are (1;-1) and (1;1).

octave:4> [V,lambda] = eig([2, 1; 1,2])
V =

  -0.70711   0.70711
   0.70711   0.70711

lambda =

Diagonal Matrix

   1   0
   0   3

Why are the eigenvectors in octave [-0.70711; 0.70711] and [0.70711; 0.70711]?

Answer 1

Given λ₁ = 3 the corresponding eigenvector is:

| 2 1 |   |x|     |x|
|     | * | | = 3 | |   =>   x = y
| 1 2 |   |y|     |y|

I.e. any vector of the form [x, x]', for any non-zero real number x, is an eigenvector. So [0.70711, 0.70711]' is an eigenvector as valid as [1, 1]'.

Octave (but also Matlab) chooses the values such that the sum of the squares of the elements of each eigenvector equals unity (eigenvectors are normalized to have a norm of 1 and are chosen to be orthogonal, to be precise).

Of course the same is valid for λ₂ = 1.