Example 17 (1998 "Hope Cup" National Invitational Competition for Senior High School Training Question) Given $x, y, z \in \mathbf{R}^{+}$, find the maximum value of $\frac{x y + y z}{x^{2} + y^{2} + z^{2}}$.

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Observe and analyze the structure of the numerator and denominator. It is only necessary to decompose \( y^2 \) such that \( y^2 = \frac{1}{2} y^2 + \frac{1}{2} y^2 \). Thus, 
\[
\frac{x y + y z}{x^2 + y^2 + z^2} = \frac{x y + y z}{\left(x^2 + \frac{1}{2} y^2\right) + \left(\frac{1}{2} y^2 + z^2\right)} \leq \frac{x y + y z}{2 \sqrt{\frac{1}{2}} x y + 2 \sqrt{\frac{1}{2}} y z} = \frac{\sqrt{2}}{2},
\]
when \( x = \frac{\sqrt{2}}{2} y = z \), the function \( \frac{x y + y z}{x^2 + y^2 + z^2} \) has a maximum value of \( \frac{\sqrt{2}}{2} \).