Writing a matrix with imaginary eigenvalues in a block-diagonal form

Question

Trying to write a 4x4 matrix $C$ in a block diagonal form $A$ where $C=QAQ^{-1}$ is valid and I am also supposed to find $Q$. I thought I could reach this in a similar fashion to finding Jordan form through using eigenvalues and eigenspaces.

However, the matrix $C$ has the characteristic polynomial ${\lambda}^4-1=0$ which leads to the eigenvalues -1, 1 but also $-i$ and $i$. I am assured that the characteristic polynomial is correct. This is given as a hint in the question.

Unfortunately, I am not getting any feedback from prof or assistants other than that I am not supposed to get on with imaginary numbers (Despite this not being mentioned anywhere in the question). So I am stumped on how to move forward. I am refraining from adding the matrix here since I do not want a solution to the question I'm supposed to solve myself, rather I'd like to know how to proceed further or at least a similar example. At this point, the lack of useful help from my course is driving me crazy. Excuse my frustration. Any help is appreciated.

Answer 1

Not a complete solution, but a hint

Hint: your block diagonal form will look something like $$ \pmatrix{ 1 & & & \\ & -1 & & \\ & & & 1\\ & & -1 & } $$ with the second block being the part corresponding to the complex eigenvalues.

It might help to look at the $2 \times 2$ matrix $M = \pmatrix{ 0 & 1 \\ -1 & 0}$ and compare what you get from diagonalizing it (the matrix $Q$ has complex entries; after diagonalization, you get a diagonal matrix with $\pm i$ on the diagonal) and what you get by block-diagonalizing it (you get back $M$!). And then maybe look at $$ M = \pmatrix{c & -s \\ s & c} $$ where $c$ and $s$ are the sine and cosine of some angle like $\pi/3$, and see whether you can do something similar. By then, you'll be 90% of the way to being done.

Incidentally, there's a cheap trick that's perfectly valid: you could say that $A = C$ and $Q = I$ represents $C$ as a block diagonal matrix with exactly one $4 \times 4$ block. But it's pretty clear that this isn't what your teacher really wants. (I might suggest considering finding a different teacher, or taking the course a different year, if that's an option...)

Answer 2

1. You have to calculate the eigenvectors, e.g. $v_1$, $v_2$, $v_3$ and $v_4$. Let the last two be the complex eigenvectors.

2. Calculate $w_1 := \frac{1}{\lVert v_1 \rVert}v_1$, $w_2 := \frac{1}{\lVert v_2 \rVert}v_2$ and $w_3 := \frac{1}{\lVert v_3 \rVert}v_3$.

3. Then $Q := \left(w_1, w_2, \frac{1}{\lVert \mathrm{Re}(w_3)\rVert}\mathrm{Re}(w_3), \frac{1}{\lVert \mathrm{Im}(w_3)\rVert}\mathrm{Im}(w_3) \right) \in \mathbb{R}^{4 \times 4}$ should do the job.