Unitarily

Unitarily is an adverb used in mathematics, physics, and related disciplines to describe actions performed by means of a unitary transformation or in a unitary manner. In linear algebra over the complex field, a unitary transformation U preserves inner products: U* U = I, where U* is the conjugate transpose of U. Equivalently, U preserves norms and angles between vectors, making it reversible.

In matrix terms, two matrices A and B are unitarily similar if there exists a unitary matrix U with A = U B U*, or B = U* A U. If a matrix is unitarily similar to a diagonal matrix, it is unitarily diagonalizable; this occurs for normal matrices, where A* A = A A*. Unitarily invariant properties, such as eigenvalue magnitudes and the Frobenius norm, remain unchanged under unitary conjugation.

In physics, unitary evolution describes the time evolution of quantum states: the evolution operator U(t) is unitary, ensuring conservation of probability. More broadly, unitary transformations are valued for preserving inner products, norms, and overall structure in numerical computations and signal processing. They are favored for numerical stability because they do not amplify errors or distort essential relationships.

Terminology: unitary is the adjective describing objects with the unitary property; unitary transformations or operators are described as acting "unitarily," i.e., in a unitary manner. The term derives from the concept of unity or wholeness in the mathematical sense of preserving structure under transformation.