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I'm trying to follow along with this: https://dandavison.github.io/fibonacci-eigenbasis.html

And I'm getting stuck on what's presumably very basic algebra. The step that's giving me trouble is going from: $$ \frac{-1}{2\sqrt5} \begin{bmatrix} -5-\sqrt5 & 0 \\ 0 & 5-\sqrt5 \\ \end{bmatrix} $$ to: $$ \frac{1}{2} \begin{bmatrix} 1+\sqrt5 & 0 \\ 0 & 1-\sqrt5 \\ \end{bmatrix} $$

the closest i can get is: $$ \frac{1}{2} \begin{bmatrix} 5(1+\sqrt5) & 0 \\ 0 & 5(1-\sqrt5) \\ \end{bmatrix} $$ and I don't understand how the extra factor of 5 comes out. What am I missing here?

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The author distributed the $-\frac{1}{\sqrt{5}}$ into each component of the matrix. This gives us $$-\frac{1}{\sqrt{5}}(-5-\sqrt{5}) = \frac{5}{\sqrt{5}}+\frac{\sqrt{5}}{\sqrt5} = \sqrt5+1$$ and $$-\frac{1}{\sqrt{5}}(5-\sqrt{5}) = -\frac{5}{\sqrt{5}}+\frac{\sqrt{5}}{\sqrt5} = -\sqrt5+1$$

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    Thank you for your answer florence. I see now I was trying to treat it like balancing an equation rather than distributing terms inside the matrix (i.e. I was trying to multiply by $\sqrt5$ rather than divide). It also wasn't intuitively obvious to me that $\frac{5}{\sqrt5}$ would equal $\sqrt5$. Seems I am rustier than I first though. Thanks again! – 2017-02-12