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- 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
- linear algebra - Singular value decomposition for unitary matrices . . .
I know the title is strange, but there are many instances in quantum information in which one is interested not in diagonalizing a unitary matrix, but instead in finding its singular value decompos
- prove that an operator is unitary - Mathematics Stack Exchange
prove that an operator is unitary Ask Question Asked 5 years, 5 months ago Modified 5 years, 5 months ago
- Operator - Exponential form - Mathematics Stack Exchange
A unitary operator is a diagonalizable operator whose eigenvalues all have unit norm If we switch into the eigenvector basis of U, we get a matrix like: \begin {bmatrix}e^ {ia} 0 0\\0 e^ {ib} 0\\0 0 e^ {ic}\\\end {bmatrix} which is obviously the exponential of a diagonal hermitian matrix
- On certain decomposition of unitary symmetric matrices
On certain decomposition of unitary symmetric matrices Ask Question Asked 13 years, 4 months ago Modified 11 years, 11 months ago
- 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
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