Suppose $W$ is given by span$\{(1,2,2)^T,(5,4,-2)^T\}$ over $\mathbb R$. Find an orthogonal change of co-ordinates matrix C $\in$ GL3 $(\mathbb R)$ and a matrix $R$ in $M_{3,3}(\mathbb R) $ such that $CRC^{-1}$ is the rotation matrix about the axis $W^c$ through an angle of $\pi$, where $W^c$ refers to the orthogonal complement of $W$.

Observations $\\$ Since the matrix is rotational about $W^c$, I know that it means that $CRC^{-1}$ has an eigenvector $w^c$ corresponding to $1$ (as it would be unchanged by the rotation). I also know that I can find two orthogonal vectors in the plane $W$, and that they should be eigenvectors of $CRC^{-1}$ with eigenvalues of $-1$ (since they would be rotated by $180$ degrees). So using these ideas, I think I can construct a matrix $B = CRC^{-1}$ which satisfies the properties.

But how can I decompose $B$ into $C$ and $R$? For any matrix I am aware that you can decompose it into a Jordan form $PJP^{-1}$, but the columns of $P$ don't necessarily have to be orthogonal (just linearly independent). I am thinking about using QR factorisation to decompose $P$ further, but it seems that all of this would take an unbelievably more amount of work than what was intended in the question.

Could someone guide me to perhaps another thought process that would be computationally easier in solving the question? Even better, if there's something wrong with my current thinking, please point them out.


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