# Matrix Groups: Quaternion Algebra

Hi I am very stuck on this problem and appreciate any form of help I can get:

Question: Verify that $M_n(\mathbb{H})$ is complete with respect to the norm $\lVert\ \rVert$. Use this to define an exponential function $\text{exp}: \text{M}_n(\mathbb{H}) \rightarrow \text{GL}_n(\mathbb{H})$ with properties analogous to those for the exponential functions on $\text{M}_n(\mathbb{R})$ and $\text{M}_n(\mathbb{C})$.

My problem is that I do not really know how to put a bound on $\lVert A\rVert = sup\{\frac{|Ax|}{|x|}:\ 0 \neq x \in \mathbb{H}^n\}$ where $A \in \text{M}_n(\mathbb{H})$, since I am not sure what is the set of basis in $\mathbb{H}^n$. Also, for the exponential function, I don't understand what I need to do to "define" the function... is it defined by the power series in this case just like $\text{M}_n(\mathbb{R})$ and $\text{M}_n(\mathbb{C})$?

• what norm? There are lots of norms on matrix rings and (while they should all work) we whould know the one you have in mind to use. May 8, 2017 at 19:02
• Yes I should have mentioned it. It is $\Lvert A\Rvert = sup\{\frac{|Ax|}{|x|}:\ 0 \neq x \in \mathbb{H}^n\} May 9, 2017 at 1:44 • Presumably the analogous properties are just$\exp(A+B)=\exp(A)\exp(B)$when$A,B$commute, or that$X(t)=\exp(tA)$satisfies$X'=AX$(as a function of real$t$)? And then, proving the properties requires justifying convergence with respect to the norm as$\exp$is defined by the power series. Also, presumably the operator norm matches the Frobenius norm, as for$\Bbb R$or$\Bbb C\$. Oh, and be aware you can edit/delete comments until you figure out how to display them properly, so you don't have to leave a comment with unrendered LaTeX hanging like you did. May 9, 2017 at 2:21