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# Unitary matrices preserve inner products.

Put content here**Theorem.** Unitary matrices preserve the standard inner product on $\mathbb{C}^n$. For any vectors $x, y \in \mathbb{C}^n$ and unitary matrix $U$:
$$\langle Ux, Uy \rangle = \langle x, y \rangle$$
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**Proof.**
$$\langle Ux, Uy \rangle = (Uy)^* (Ux) = y^* U^* U x = y^* x = \langle x, y \rangle$$
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**Consequences:**
- Norm preservation: $\|Ux\| = \|x\|$ (unitary matrices are isometries)
- Distance preservation: $\|Ux - Uy\| = \|x - y\|$ (they are rigid motions)
- Angle preservation: the angle between $Ux$ and $Uy$ equals the angle between $x$ and $y$
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This makes unitary transformations the natural notion of "rotation" in complex vector spaces.

# Parents

* Unitary matrices