How do unitary matrices preserve the magnitude of unit vectors?

Question

In quantum mechanics and quantum computing, quantum particles evolve in a unitary manner. That is to say at any point in time the particle/system (represented as a vector) has a magnitude of 1, meaning that all the different probabilities of its states sum to 100%.

This is all fine and well, but why is it that only matrices that are their own inverse (i.e unitary matrices) satisfy this property? Do they preserve the magnitude all vectors? or just unit vectors?

EDIT: My bad unitary matrices are the inverse of their own Hermitian conjugate.

Answer 1

No, a unitary matrix is not its own inverse. It is the inverse of its Hermitian conjugate.

Using the polarization identity, a matrix preserves magnitudes of vectors (all vectors, not just unit vectors) if and only if it preserves the inner product, and that is true if and only if the matrix is unitary:

$$ \langle U x, U y \rangle = \langle x, U^* U y \rangle = \langle x, y \rangle \ \text{for all} \; x,y \; \text{iff}\ \ U^* U = I$$

Answer 2

Unitary matrices are invariant under the 2 norm. You can show this like the following.

$$\| Qx\|_{2}^{2} = \|x\|_{2}$$

$$ \| Qx \|_{2}^{2} = (Qx)^{*}Qx $$ $$ =x^{*}Q^{*}Qx $$ $$ =x^{*}\underbrace{Q^{-1}Q}_{I}x $$ $$ =x^{*}x $$ $$ =\|x\|_{2}^{2} $$