Eigenvectors and eigenvalues

The directions the matrix does not deflect

A matrix deforms space: it rotates, stretches, shears. But some directions escape the rotation — they are merely stretched. These privileged directions are the eigenvectors, and they reveal the deep nature of the transformation.

The definition

A non-zero vector v is an eigenvector of the square matrix A if there is a number λ such that:

A v = λ v

The number λ is the associated eigenvalue. In other words: applying A to v does not change its direction, only its length (and possibly its sense, if λ < 0).

    an arbitrary vector             an eigenvector
         ^                               ^
         |   A                           |   A
      u  |  ---->  Au                 v  |  ---->  Av = 2v
        /              \                 |            |
       /                \                |            |
      O                  O               O            O

  the direction changes          the direction is preserved

The zero vector is excluded: A·0 = λ·0 would hold for every λ, which says nothing. On the other hand λ = 0 is allowed: it means v is crushed, that is, v lies in the kernel.

The geometric examples

They make the notion obvious:

Transformation                Eigenvectors              Eigenvalues
----------------------------  ------------------------  ---------------
scaling by 3                  every vector              3
projection onto the x-axis    the x-axis -> kept        1
                              the y-axis -> crushed     0
reflection in the x-axis      the x-axis -> unchanged   1
                              the y-axis -> flipped    -1
90° rotation in the plane     none (real)               none (real)

The last line is instructive: a quarter-turn rotation deflects every direction, so it has no real eigenvector. (It does have complex ones, i and -i — one of the links between the two halves of this course.)

The eigenspace

If v is an eigenvector for λ, then so are 2v, -v and all its multiples:

A(2v) = 2(Av) = 2(λv) = λ(2v)

The set of vectors satisfying Av = λv, together with the zero vector, therefore forms a vector subspace called the eigenspace:

E(λ) = Ker(A - λI)

This formulation is crucial: finding the eigenvectors of λ means solving the homogeneous system (A - λI)X = 0. The whole previous course pays off here.

What these directions mean

An eigenvector is a stable direction: a state the transformation does not mix with the others. Depending on the context, it is called a normal mode of vibration (mechanics), a stationary state (probability), a principal axis of inertia (physics) or a principal component (statistics). It is always the same equation.

Summary