A Unitary matrix is defined as a square matrix whose conjugate transpose is also its inverse.

More formally, a complex square matrix $U$ is considered unitary if it satisfies the condition:

where $U^H$ denotes the conjugate transpose of $U$, and $I$ represents the identity matrix. This property implies that a unitary matrix preserves the inner product of vectors. Consequently, when a unitary matrix is applied to a vector, both the length of that vector and the angle it forms with other vectors in the space remain unchanged.

Unitary matrices possess several important characteristics. For instance, the columns of a unitary matrix form an orthonormal basis for the corresponding vector space. This indicates that all columns are unit vectors (having a length of 1) and are mutually orthogonal. Additionally, another significant property is that the determinant of a unitary matrix has an absolute value of 1, which implies that a unitary matrix does not alter the volume of the space it operates within.

Unitary matrices find applications in various fields of mathematics and physics, including quantum mechanics and signal processing. In quantum mechanics, unitary matrices are utilized to represent the evolution of quantum states over time. In signal processing, they play a crucial role in data compression and encryption techniques.