What is unitary transformation?

A unitary transformation is a type of linear transformation in which the inner product of vectors is preserved. It is often used in quantum mechanics and signal processing to describe transformations that preserve the norm of a vector.

In mathematics, a unitary transformation is a linear transformation that preserves the inner product of vectors in a complex vector space. This means that for any two vectors u and v, the inner product <u, v> is equal to <Tu, Tv>, where T is the unitary transformation matrix.

Unitary transformations are important in quantum mechanics because they preserve the probability amplitudes of quantum states. In signal processing, they are used to describe transformations that preserve the energy of a signal.

Unitary transformations are also commonly used in the theory of group representations, where they provide a way to transform between different representations of a group while preserving certain properties.

Overall, unitary transformations play a key role in many areas of mathematics and physics, providing a way to understand and describe transformations that preserve important properties of vectors and signals.