I am trying to understand the intuition behind eigenvalues/eigenvectors through the lens of repeated matrix multiplication:
Given a $2\times2$ matrix $M$ and $2D$ vector $v$, multiplying $v$ repeatedly
with $M$ causes the result ($M^n v$) to gravitate towards one of the eigenspaces of $M$ because:
$$M^n v = M^n(\alpha x_1 + \beta x_2) = (\alpha \lambda_1^n x_1 + \beta \lambda_2^n x_2)$$
where $x_1$ and $x_2$ are eigenvectors of $M$ and $\lambda_1$ and $\lambda_2$ the corresponding eigenvalues. As $n$ gets larger $M^n v$ will gravitate towards either $\alpha \lambda_1^n x_1$ or $\beta \lambda_2^n x_2$, whichever has the dominant eigenvalue.
assuming: $v = \alpha x_1 + \beta x_2$
So the above is a way to connect the abstract concept of eigenvalue/eigenvector to something concrete: what happens when you apply a matrix over and over to a vector.
However, the intuition breaks down for me with complex eigenvectors. I know repeated multiplication by a matrix with complex eigenvectors causes the result to either spiral outwards or inwards.
Is there simple math such as above to see why?
Edit: I know similar questions have been asked before, but I ask in the context of repeated matrix multiplication
Best Answer
Every square matrix is similar to a matrix in what's called Jordan Canonical Form. This has various properties, but most important here is that it is upper triangular, and the eigenvalues (of both the new and original matrix) are on the diagonal of the resulting matrix.
The way to think about this process is that we change bases, and in that new basis the matrix becomes diagonal. This will help the existing intuition given in the OP about iterating the matrix, because iterating a triangular matrix will simply exponentiate the diagonal entries (and do predictable but somewhat messy stuff to the part above the diagonal).
Now, if an eigenvalue is complex, all the above still holds. However if the original matrix had real entries, then that eigenvalue must be paired with its complex conjugate -- both $a+bi$ and $a-bi$ must be eigenvalues. Then, we can rearrange the standard JCF matrix, as described above, into what is called Real Jordan Canonical Form. Now, instead of the matrix being entirely upper triangular, there will be some $2\times 2$ blocks along the diagonal, a rearrangement of what was a complex eigenvalue-plus-its-conjugate pair in the original JCF. The entries of these $2\times 2$ blocks are exactly the real and imaginary parts of these two complex eigenvalues.
Each of these $2\times 2$ blocks performs a rotation in the two-dimensional space spanned by those two basis elements. Hence, a $6\times 6$ matrix with six complex eigenvalues might be doing three different rotations in three different two-dimensional directions at once.
In the original $2\times 2$ case, the reason that a complex eigenvalue leads to a rotation is that it must appear with its complex conjugate, assuming the original matrix has real entries. As a pair they give a rotation. If the original matrix did not have real entries, then the matrix need not represent a pure rotation, because the two eigenvalues need not be related.