# Logical Explanation behind second reflection in Grover's algorithm

Logically, what does the second reflection in Grover's algorithm do? I currently don't have a lot of background in linear algebra so I'm failing to understand what exactly its purpose is and what it does.

An application of the reflection operator change each amplitude of a basis state $$|i\rangle$$ by $$\alpha_i \rightarrow- \alpha_i + 2 \langle\alpha\rangle\$$ where $$\langle\alpha\rangle\$$ is the average of all amplitudes. It follows the oracle which is use to "mark" the seeked elements $$|i\rangle$$.
Say for instance that you have 8 elements, so you work on 3 qubits and you want to output $$|011\rangle$$ with high probability. At the beginning, each basis state has $$\frac{1}{8}$$ probability to be measured as their amplitude are $$\frac{1}{\sqrt{8}}$$. The oracle first will mark $$|011\rangle$$, so its amplitude will be $$-\frac{1}{\sqrt{8}} = -\frac{1}{2\sqrt{2}}$$.
$$\langle\alpha\rangle\ = \frac{1}{8} (7* \frac{1}{\sqrt{8}} - \frac{1}{\sqrt{8}}) = \frac{6}{8\sqrt{8}} = \frac{3}{8\sqrt{2}}$$
So the amplitude of $$|011\rangle$$ after the reflection operator becomes $$- (-\frac{1}{2\sqrt{2}}) + 2\langle\alpha\rangle\ = \frac{5}{4\sqrt{2}}$$ while others will have a new amplitude: $$- (\frac{1}{2\sqrt{2}}) + 2*\frac{3}{8\sqrt{2}} = \frac{1}{4\sqrt{2}}$$
If you square those amplitudes, then $$|011\rangle$$ has $$\frac{25}{32} \approx 0.78 > \frac{1}{8}$$ probability to be measured, while another state will have $$\frac{1}{32}$$ probability to be measured. You see the probability of the solution to be outputed increased while non-solutions less likely to be.