Definition: A complex square matrix $U$ is unitary if its conjugate transpose equals its inverse:
$$U^* U = UU^* = I \quad \text{or equivalently} \quad U^{-1} = U^*$$
where $U^* = (\overline{U})^T$ is the conjugate transpose.
Example:
$$U = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$$
Properties:
- $|\det(U)| = 1$
- Preserves inner products: $\langle Ux, Uy \rangle = \langle x, y \rangle$
- Preserves norms: $\|Ux\| = \|x\|$
- Eigenvalues lie on the unit circle ($|\lambda| = 1$)
- Columns (and rows) form an orthonormal basis
- The product of unitary matrices is unitary
Unitary matrices are the complex analog of orthogonal matrices. They represent isometries (distance-preserving transformations) in complex vector spaces and are fundamental in quantum mechanics.
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- Definition of unitary matrixDefinition. A complex square matrix $U$ is unitary if its conjugate transpose equals its inverse: $$U^ U = U U^ = I$$...
- Unitary matrices are invertible.Fact. Every unitary matrix is invertible. By definition, if $U$ is unitary then $U^ U = I$, which means $U^{-1} = U^...
- Unitary matrices have orthogonal (orthonormal) rows/columns.A complex square matrix $U$ is unitary iff $U^ U=I$, equivalently iff its columns form an orthonormal basis. Using...
- Unitary matrices preserve inner products.For a unitary matrix $U$, $\langle Ux,Uy\rangle=(Ux)^ (Uy)=x^ U^ Uy=x^ y=\langle x,y\rangle$. Thus unitary...
- Unitary matrices preserve orthogonal (orthonormal) bases.Theorem. If $\{v 1, v 2, \ldots, v n\}$ is an orthonormal basis of $\mathbb{C}^n$ and $U$ is a unitary matrix, then...
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