Tagged Questions

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Matrix multiplication in quaternions is not necessarily linear

I tried to show by example that matrix multiplication for quaternionic matrices is is not necessarily $\mathbb H$-linear. If $A \in M_n(\mathbb H)$ is a quaternionic matrix and $x$ is a vector in ...
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Why not $SL_n (\mathbb R)$ in this exercise

I just solved the following exercise: Let $SL_2(\mathbb Z)$ denote the set of $2\times2$ matrices with integer entries and determinant $1$. Prove that $SL_2(\mathbb Z)$ is a subgroup of ...
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Is this a tangent bundle and what is the meaning of this exercise

I intend to solve the following exercise but I would like to have some help with understanding the ''big picture'': Exercise. Describe a natural 1 to 1 correspondence between elements of $SO(3)$ ...
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Left-invariant Riemannian metric on $SO(3)$

Let's consier the manifold $SO(3)$. First problem is to show that $T_I SO(3)$ is a space of skew-symmetric matrices $3\times 3$. How can I deduce it? Then I have to prove there exists exactly one ...
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Group operation in $SO(3)$ is well-defined

I've just started to read Tapp's Matrix groups for undergraduates and it says: "$SO(3)$ becomes a group under composition of motions (since different motions might place the globe in the same position ...
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Fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n, \mathbb Z)$

What is a simple description of a fundamental domain of $\operatorname{GL}(n,\mathbb R)$ acted on by $\operatorname{GL}(n,\mathbb Z)$? $\operatorname{GL}(n,\mathbb R)$ is the group of all real ...
Consider $$T_{2}: \left[ \begin{array}{cc} a & b \\ c & d \\ \end{array} \right] \rightarrow \left[ \begin{array}{cc} d & c \\ b & a \\ \end{array} \right]$$ $$T_{3}: \left[ ... 4answers 349 views How to show path-connectedness of GL(n,\mathbb{C}) Well, I am not getting any hint how to show GL_n(\mathbb{C}) is path connected. So far I have thought that let A be any invertible complex matrix and I be the idenity matrix, I was trying to ... 1answer 394 views What is the Lie algebra of the indefinite orthogonal group''? For p,q \geq 0 and n=p+q\geq 1, give \mathbb{R}^n the indefinite inner product (written as a matrix)$$ \begin{pmatrix} I_p & \\ & -I_q \end{pmatrix},  where $I_m$ is the $m \times m$ ...
Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices: Cayley's Transformation with ...