What is the difference between isometric and unitary operators on a . . . A stronger notion is unitary equivalence, i e , similarity induced by a unitary transformation (since these are the isometric isomorphisms of Hilbert space), which again cannot happen between a nonunitary isometry and a unitary operator (or between any nonunitary operator and a unitary operator)
linear algebra - Norm preservation properties of a unitary matrix . . . Definition (Unitary matrix) A unitary matrix is a square matrix $\mathbf {U} \in \mathbb {K}^ {n \times n}$ such that \begin {equation} \mathbf {U}^* \mathbf {U} = \mathbf {I} = \mathbf {U} \mathbf {U}^* \end {equation} Definition (Vector $2$ -norm)
linear algebra - Whats the interpretation of a unitary matrix . . . Unitary matrices are the complex versions, and they are the matrix representations of linear maps on complex vector spaces that preserve "complex distances" If you have a complex vector space then instead of using the scaler product like you would in a real vector space, you use the Hermitian product
If H is Hermitian, show that $e^ {iH}$ is unitary In the case where H is acting on a finite dimensional vector space, you can essentially view it as a matrix, in which case (by for example the BCH formula) the relation you state in a) is valid More generally if $ [A,B]=0$ then the product of exponentials is just the exponential of the sum There may be subtleties in the more general case, but I doubt you'd even be interested in those As for
Prove the Fourier transform is a unitary linear operator I am trying to prove that the inverse of the fourier transform is equal to its adjoint (i e it is a unitary linear operator) I am working with the inner product $\langle s_1,s_2 \rangle=\int_ {-\