What Is a Unitary Matrix?

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A unitary matrix is a matrix that when multiplied by its complex conjugate transpose matrix, equals the identity matrix. This implies that the complex conjugate transpose of a matrix is equal to the inverse of the unitary matrix. Unitary matrices have several applications in different fields of science and engineering, such as quantum mechanics.

Unitary matrices with entries that are all real are orthogonal matrices. The rows of unitary matrices are a unitary basis. This means that each row has length one and their Hermitian inner product equals zero. Various linear algebra techniques can be used to determine if a matrix is unitary.