UNITARY MATRIX Meaning and
Definition
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A unitary matrix, in the field of linear algebra, refers to a square matrix that maintains certain significant properties under multiplication. Specifically, a unitary matrix is an n x n matrix U, where n represents the size of the matrix, that possesses the following two criteria:
1. Orthogonality: The columns and rows of a unitary matrix are orthonormal, which means they form a set of mutually orthogonal vectors. In other words, the inner product of any two columns or rows is zero if they are distinct, and one if they are the same.
2. Conservation of Norm: The norm, or length, of any vector is preserved after multiplying it by a unitary matrix. Thus, if a vector v is multiplied by the unitary matrix U, the norm of the resulting product Uv remains the same as the norm of v.
These properties make unitary matrices fundamental in several areas of mathematics and physics, including quantum mechanics and signal processing. In quantum mechanics, unitary matrices correspond to transformations that preserve the probabilistic interpretations of quantum states. In signal processing, unitary matrices play a central role in operations such as Fourier analysis and wavelet transforms.
The unitary matrix is also closely related to the concept of the adjoint or conjugate transpose. Taking the adjoint of a unitary matrix results in its inverse, emphasizing the crucial role unitary matrices play in preserving orthogonality and norm.
Etymology of UNITARY MATRIX
The word "unitary" in mathematics is derived from the Latin word "unitas", meaning "oneness" or "unity". It refers to a property or concept that involves unity or preserving a sense of oneness.
In the context of linear algebra and matrices, a unitary matrix is a square matrix whose conjugate transpose (also known as the Hermitian adjoint) is equal to its inverse. The term "unitary" is used to describe this property because it implies that the matrix maintains a sense of unity or oneness when undergoing conjugate transposition.
Overall, the word "unitary" in "unitary matrix" is used to convey the concept of preserving unity or oneness under certain transformations.
