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Finding $U,V$ in Thompson's Formula

Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that: $e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$ Given $a,b \in \mathfrak{su}(4)$ defined by: $a=J_x ...
Benjamin's user avatar
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4 votes
0 answers
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How to find the unitary matrices in this exponential matrix representation

In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials: if $A$ and $...
Lababidi's user avatar
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3 votes
0 answers
142 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
André Schlichting's user avatar
3 votes
0 answers
83 views

Particular decomposition of $SU(n)$

Given $a,b \in \mathfrak{su(n)}$ which generate the full algebra, it is possible to write and $G \in SU(n)$ as: $G = \exp(\alpha_1 a)\exp(\beta_1 b) \ldots \exp(\alpha_m a)\exp(\beta_m b)$ for some ...
Benjamin's user avatar
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3 votes
0 answers
112 views

Find logarithm of a matrix containing a constrained set of basis elements

Let $U$ be a unitary matrix, and let $H$ be an Hermitian matrix. I want to know if there is a $t \in\mathbb R$ such that $\exp(i t H) = U$. A connected question is: given a set $\{g_1, g_2, ..., g_N\}...
glS's user avatar
  • 342
2 votes
0 answers
329 views

Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$? Is it enough to consider the injection via Hochschild: $Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$? Edit: The ...
cheyne's user avatar
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2 votes
0 answers
90 views

Singularities of the Quantum propagator (baby version)

Given $a,b \in \mathfrak{su}(4)$ which are taken to generate the whole algebra, consider the following map $V:\mathbb{R}^{2} \rightarrow SU(4)$: $V : (w_1, w_{2}) \mapsto e^{(a+w_2 b)} e^{(a+w_1 b)}$ ...
Benjamin's user avatar
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