Concept# Unitary matrix

Summary

In linear algebra, an invertible complex square matrix U is unitary if its conjugate transpose U* is also its inverse, that is, if
U^* U = UU^* = UU^{-1} = I,
where I is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is referred to as the Hermitian adjoint of a matrix and is denoted by a dagger (†), so the equation above is written
U^\dagger U = UU^\dagger = I.
For real numbers, the analogue of a unitary matrix is an orthogonal matrix. Unitary matrices have significant importance in quantum mechanics because they preserve norms, and thus, probability amplitudes.
Properties
For any unitary matrix U of finite size, the following hold:

- Given two complex vectors x and y, multiplication by U preserves their inner product; that is, ⟨Ux, Uy⟩ = ⟨x, y⟩.
- U is normal (U^* U = UU^*).
- U is d

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2004