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Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...
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$SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$. I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...
Let $V$ be the space of $4$ by $4$ Hermitian matrices, that is a vector space of dimension $16$ over $\mathbb{R}$. Is the uniform measure of $$\left\{ W\in Gr\left(5,V\right):W \text{contains no ... 1answer 469 views A question on eigenvalues Let A_{1}, A_{2}, A_{3}, A_{4}, A_{5} be linearly independent Hermitian matrices in the the space of 6 by 6 Hermitian matrices as a vector space over \mathbb{R}. Does there always ... 1answer 118 views On matrices conjugated in a faithful representation Let k an algebraically closed field. Let O=k[[\pi]] and F=k((\pi)) and G\rightarrow GL_{n} a faithful representation of a semisimple group. Let A, B\in G(O)\cap G(F)^{rs} (rs for regular ... 0answers 79 views integral stable conjugacy classes Let G be a semisimple simply connected group over k algebraically closed field . Let \gamma,\gamma'\in G(k[[\pi]]) that are generically regular semisimple on G(k((\pi))). We assume that ... 0answers 151 views How to find the unitary matrices in this exponential matrix representation In the following post Representing a product of matrix exponentials as the exponential of a sum there is a statement regarding the result of the multiplication of two matrix exponentials: if A and ... 0answers 131 views Compute the discriminant for reductive groups Consider G=GL_{2} and F=k((\pi)), and a diagonal matrix t=\left(\begin{array}{cc}a&0\\0&b\end{array}\right). The characteristic polynomial of t is X^{2}-(a+b)X+ab, and the ... 1answer 285 views Iwasawa Decomposition for Matrices [closed] I was asked to prove that if$$ T_{n}^{+}(\mathbb{R}) \subseteq M_{n}(\mathbb{R})$$denotes the set of upper triangular matrices with positive diagonal entries, then prove that the multiplication ... 1answer 224 views identifying dual of lie algebra of general linear groups Is there any reference for the following fact? I am looking for a nice and simple proof. Assume that G=GL(n,C), the group of invertible n\times n matrices with complex entries. Why can the dual ... 2answers 316 views Have derivatives of determinants along 1-psg's ever been 'coherently' computed via Jacobi's formula? Suppose \mathfrak{p} denotes all the symmetric matrices in \mathfrak{sl}_{2n} \mathbb{R}. Then for each parameterized 1-dimensional linear subspace \xi=\xi(t) of \mathfrak{p} we get a ... 3answers 370 views Matrices generating non-discrete subgroups of SL(2,R) Jorgensen's inequality \mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1 gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ... 3answers 317 views Linear subspaces in cones over orthogonal groups Consider the orthogonal group G=O(n) as a subset of the vector space of n\times n real matrices. Let C=C(G) denote the Euclidean cone over G, i.e., the space of matrices of the form tA, A\in ... 0answers 308 views Spaces of matrices with same eigenvalue/Great circles in O(n)-orbits Let Sym^2(V) be the set of symmetric matrices of a real n-dimensional vector space V. Given an element \underline{\lambda}=[\lambda_1,\ldots \lambda_n]\in \mathbb{RP}^n, where ... 4answers 1k views Representing a product of matrix exponentials as the exponential of a sum In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if A and B are Hermitian matrices, then there exist unitary matrices ... 1answer 308 views Invariants of co-diagonalizability in real symmetric matrices This question has been mentionned to me by U. Frisch. He wanders whether it has ever been considered by algebraists. In the vector space {\bf Sym}_n({\mathbb R}), two elements commutte to each ... 1answer 225 views omega-Commuting matrices vs Stone-von Neumann Theorem Let me first recall the Stone-von Neumann theorem that if two one-parameter groups of unitary operators U_t and V_s over a Hilbert space satisfy U_tV_s=e^{ist}V_sU_t for every s,t\in{\mathbb ... 0answers 164 views criterion for deciding whether the product of a sequence of Givens rotations can reach the full special orthogonal group By Givens' rotation R(1,2;\theta) I mean a matrix which has the$$\begin{pmatrix} \hphantom{-}\cos \theta &\sin \theta \cr -\sin \theta & \cos \theta \end{pmatrix}$$2 \times 2 block at ... 2answers 663 views What's the Lipschitz constant of the exponential map for SO(n,R)? Consider the Lie algebra so(n) equipped with the metric \langle e_i \wedge e_j, e_k \wedge e_l \rangle = \delta_{i,k} \delta_{j,l}. Similarly equip the tangent space at other points of SO(n) by ... 1answer 341 views Analogies between orthogonal/unitary groups and their indefinite counterparts Suppose I have A\in U(n) such that A^t=A (which is a bit un-natural, as usually you'd consider the hermitian transpose, not the transpose). Well, then A=X+iY say, for X and Y real matrices. ... 4answers 991 views Parametrization of O(3) Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices? 2answers 2k views Left and right eigenvalues A quaternionic matrix A gives rise to a function \mathbb{H}^n \to \mathbb{H}^n given by x \mapsto A \cdot x. This is real linear, but not complex- or quaternionic-linear (in general) if we ... 1answer 455 views What matrix groups can be embedded in Sp_4? In a joint paper with Yifan Yang we constructed an "exotic" embedding of SL_2(\mathbb R) in Sp_4(\mathbb R) (in fact, of PSL_2(\mathbb R) in PSp_4(\mathbb R)), namely,$$ ...
Hi, I've been looking for a clear reference which shows that the matrix exponential is surjective from $M_{n}(C)$ to $Gl_{n}(C)$. Wikipedia claims this is true, but I haven't seen it proven... ...