Unitary Transformation

a linear transformation

with complex coefficients that leaves invariant the sum of the squares of the absolute values of the transformed quantities

Since it preserves the length

of a vector x with components x₁,x₂, . . ., x_n, a unitary transformation is the extension to a complex n-dimensional vector space of the notion of a rotation in the Euclidean plane or in Euclidean 3-space. The coefficients of a unitary transformation form a unitary matrix. The unitary transformations of an n-dimensional complex space form a group under multiplication. If the coefficients u_ij and the transformed quantities x_i are real, then the unitary transformation reduces to an orthogonal transformation of an n-dimensional real vector space.