Unitary matrices in quantum mechanics constitute a class of linear operators that preserve the inner product and total probability amplitude within Hilbert spaces. These transformations represent time evolution or state rotations where the adjoint inverse equals the transpose, ensuring the preservation of unit norm for all vectors under transformation. As fundamental components of finite-dimensional vector space theory applied to physical states, they define reversible dynamics strictly bounded by conservation laws inherent to closed quantum systems.
Explains unitary operators as norm-preserving transformations satisfying U-dagger U = I and why quantum time evolution is unitary.