Example 5 Let $x, y, z$ be 3 non-zero real numbers. Find the maximum value of $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$.

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Introducing the parameter $\beta$, by the mean inequality (here $\beta \in \mathbf{R}^{+}$, and $\beta^{2}<1$)
$$
x^{2}+\beta^{2} y^{2} \geqslant 2 \beta x y,\left(1-\beta^{2}\right) y^{2}+z^{2} \geqslant 2 \sqrt{1-\beta^{2}} y z
$$

Thus, $x^{2}+y^{2}+z^{2} \geqslant 2\left(\beta x y+\sqrt{1-\beta^{2}} y z\right)$, let $\frac{\beta}{\sqrt{1-\beta^{2}}}=\frac{1}{2}$, solving gives $\beta=\frac{\sqrt{5}}{5}$
Therefore, $x^{2}+y^{2}+z^{2} \geqslant \frac{2}{5} \sqrt{5}(x y+2 y z)$, i.e., $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}} \leqslant \frac{\sqrt{5}}{2}$, and when $x=k, y=\sqrt{5} k, z=$ $2 k$ (here $k \neq 0$), the equality holds, $\frac{x y+2 y z}{x^{2}+y^{2}+z^{2}}$ takes $\frac{\sqrt{5}}{2}$, thus its maximum value is $\frac{\sqrt{5}}{2}$.