## Thursday, September 13, 2018

### Eigenv{ector;alue}s

There are some nice instructional videos on YouTube that explain eigenvectors and eigenvalues. What follows is a brief summary as well as a caveat or two.

We start with a square transformation matrix M, which can have one or more eigenvectors. If you restrict eigenvalues to real numbers (i.e. if you exclude imaginary numbers) then it's possible to have a transformation matrix without any real eigenvalues (e.g. in a pure rotation). It's also possible to have just one eigenvalue but an infinite number of eigenvectors (e.g. common scale in all axes) ... but I've probably distracted you too much already...

Anyway:
What's an eigenvector? It's a vector that only changes (stretches/squeezes) in scale along its span when transformed by M.
What's a span? That's a radial line to any vector from the origin of the coordinate system.
Each eigenvector has a corresponding scalar eigenvalue λ that, when the two are multiplied, has the same result as M being multiplied by that eigenvector.
It's a rather circular definition so far:
$$\matrix{M}\times\vec{v} = λ\times\vec{v}$$
We already know about a transformation matrix that represents a uniform scale of λ, and that's $$λ\times\matrix{I}$$ (it's the identity matrix, but instead of 1s running down the diagonal, it's got λs). Now we have a starting point for our calculation:
$$\matrix{M}\times\vec{v} = (λ\times\matrix{I})\times\vec{v}$$
As tempting as it looks, we should never divide by a vector. Instead, what we want next is the value of λ that results in:
$$(\matrix{M} - (λ\times\matrix{I}))\times\vec{v} = \vec{0}$$
Let's let:
$$\matrix{T} = \matrix{M} - (λ\times\matrix{I})$$
Ordinarily, it would now be trivial to solve for the eigenvector using:
$$\matrix{T}\times\vec{v} = \vec{0}$$
$$=> \matrix{T}^{-1}\times\vec{0} = \vec{v}$$
But we cannot do that in this instance, because:
$$det(\matrix{T}) = 0$$
And that means that T is not invertible. But surely that means we've missed a step? Nobody said we need the determinant to be zero. What's happening? I'll explain: if $$\vec{v} = \vec{0}$$ then anything multiplied by $$\vec{v}$$ will also equal $$\vec{0}$$.
If we restrict our eigenvector to non-zero vectors then the only way we can get a zero vector after transformation is if the transformation matrix collapses a dimension: i.e. the determinant (or scale) of the transform must be zero. That's how we get our constraint.

Note: Matrices are either invertible or singular. A matrix is invertible iff the determinant is non-zero, and it is singular iff the determinant is zero. Our transform T must be singular. That means: it collapses down by a dimension.

So, if the determinant of our 2X2 transformation matrix is zero:
$$det\begin{bmatrix}a-λ & b \\ c & d-λ \\\end{bmatrix} = 0$$
Then we know (from the Leibniz formula of a determinant) $$(a-λ) \times (d-λ) - c \times b = 0$$
And we can expand to:
$$λ^2 - λ(a+d) + (a \times d) - (c \times b) = 0$$
That's just a quadratic, easily solved (but take care with complex numbers, i does appear in rotation matrices). So we have our eigenvalue(s) λ and (by substitution) also T. But how do we find the eigenvectors if we cannot take the inverse of T? Well, you can try a technique like row reduction. But don't expect a unique solution... in fact, because the transform has a determinant of zero, the eigenvector is strictly a ratio of X:Y in two dimensions (or X:Y:Z in three dimensions, etc.) and you're going to find the lines overlap; in jargon: they are linearly dependent.