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 …

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 …

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 …

Jan 13, 2021 · 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 …

By definition a matrix $T$ is unitary if $T^*T=I.$ For two real matrices $A,B$, the $i,j$ entry of $AB$ is the inner product of the $i$ row of $A$ and $j$ column of $B$.

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 …

Nov 7, 2016 · The result does not give me the diagonal matrix with the desired eigenvalues though. Also, Google search did not yield a single nicely explained way to do a unitary transform of a normal …

are both unitary and Hermitian (for $0 \le \theta \le 2\pi$). I call the latter type trivial, since its columns equal to plus/minus columns of the identity matrix. Do such matrices have any significance (in theory …

So I'm trying to understand why the columns of a unitary matrix form an orthonormal basis. I know it has something to do with the inner product, but I don't fully understand that either (we learned all of this …

Mar 22, 2016 · Prove that the DFT Matrix is Unitary Ask Question Asked 9 years, 9 months ago Modified 1 year, 2 months ago