How can I block diagonalise this matrix?

Question

I have this matrix:

$$A = \left( \begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ \end{array} \right)$$

and I would like to show that it can be block diagonalised into :

$$ B = \left( \begin{array}{cccc} \cos 2\pi/5 & -\sin 2\pi/5 & 0 & 0 \\ \sin 2\pi/5 & \cos 2\pi/5 & 0 & 0 \\ 0 & 0 & \cos 4\pi/5 & -\sin 4\pi/5 \\ 0 & 0 & \sin 4\pi/5 & \cos 4\pi/5\\ \end{array} \right) = \left( \begin{array}{cccc} \frac{1}{4} \left(-1+\sqrt{5}\right) & -\sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & 0 & 0 \\ \sqrt{\frac{5}{8}+\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-1+\sqrt{5}\right) & 0 & 0 \\ 0 & 0 & \frac{1}{4} \left(-1-\sqrt{5}\right) & -\sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} \\ 0 & 0 & \sqrt{\frac{5}{8}-\frac{\sqrt{5}}{8}} & \frac{1}{4} \left(-1-\sqrt{5}\right) \\ \end{array} \right)$$

What is the general procedure?

Answer 1

When you diagonalise a matrix over the complex numbers, you start by finding its eigenvalues, which may be complex. If the matrix is diagonalisable, then we can expect a basis of eigenvectors corresponding to these eigenvalues. If we compute the transformation $x \mapsto Ax$ in terms of this basis of eigenvectors, we get a complex diagonal matrix, similar to $A$.

To block-diagonalise $A$ over the reals, again find all the complex eigenvalues and a basis of eigenvectors. Suppose $\alpha + i \beta$ is a non-real eigenvalue, with corresponding eigenvector $v + i w$, where $v, w$ are vectors with real components. One can easily verify that $\alpha - i \beta$ is another eigenvalue with eigenvector $v - i w$. We can form a basis of eigenvectors such that, if $v + iw$ is in the basis (where $v$ and $w$ are real vectors), then so is $v - iw$.

Now, form a real basis like so: replace the conjugate eigenvector pairs $v \pm i w$ with the vectors $v, w$. If $v + iw$ has an eigenvalue $\alpha + i \beta$, then \begin{align*} Av &= A \left(\frac{(v + iw) + (v - iw)}{2}\right) \\ &= \frac{(\alpha + i\beta)(v + iw) + (\alpha - i\beta)(v - iw)}{2} \\ &= \frac{\alpha v + i \beta v + i \alpha w - \beta w + \alpha v - i \beta v - i \alpha w - \beta w}{2} \\ &= \alpha v - \beta w. \end{align*} Similar calculation reveals $$Aw = \beta v + \alpha w.$$ So, when computing the matrix for $x \mapsto Ax$ with this basis, you'll find a block diagonal form, where each block is either a $1 \times 1$ block containing a real eigenvalue, or a $2 \times 2$ block of the form $$\begin{pmatrix} \alpha & \beta \\ -\beta & \alpha \end{pmatrix},$$ where $\alpha \pm \beta i$ is a non-real eigenvalue of $A$.

Answer 2

it is already a companion matrix (both that form and its transpose are used, depends on circumstances). The characteristic polynomial is $$x^4 + x^3 + x^2 + x + 1 = \frac{x^5-1}{x-1}$$ With four distinct eigenvalues (complex) it diagonalizes. Next, you need to figure out how to take a specific diagonal matrix, with complex $\alpha$ such that $|\alpha| = 1,$ $$ \left( \begin{array}{rr} \alpha & 0 \\ 0 & \bar{\alpha} \end{array} \right) $$ and send it back to one of your two by two blocks with sine and cosine.

To reverse the job: with some real angle $\theta,$ exactly how do you diagonalize $$ \left( \begin{array}{rr} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \; \; ? \; $$