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The matrices $A\in \mathbb{R}^{n \times n}$ and $B=\mathrm{exp}(-Q)\;A\;\mathrm{exp}(Q),$ where $Q$ is any matrix in $\mathbb{R}^{n \times n}$, are clearly similar. My questions are:

  1. Does the union of all $B,$ when we let $Q$ vary freely, $$\bigcup\limits_{Q\in \mathbb{R}^{n \times n}} B,$$ form a continuous path in $\mathbb{R}^{n \times n}$? Can we somehow specify it, or perhaps even give a parameterization in terms of $A$?
  2. Does the union above contain all matrices similar to $A$? I guess an equivalent question would be if there exists an surjection from $\{\mathrm{exp}(Q)$ : $Q\in \mathbb{R}^{n \times n}\}$ to $\mathrm{GL}_n(\mathbb{R})$.

This area of mathematics is new territory for me, so I unfortunately don't have many tools that would allow me to have a go at these problems. I'm inquiring about this because I hope to use it for my bachelor's thesis (in physics). I hope the questions are clear though. Thank you.

Bobson Dugnutt
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1 Answers1

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I will answer my own questions, except for the second part of the first question ("Can we somehow specify it, or perhaps even give a parameterization in terms of $A$?") - I would still like to hear inputs regarding this.

  1. Answering whether or not the union forms a cont. path is actually as simple as answering whether or not the function $B:\mathbb{R}^{n \times n}\rightarrow \mathbb{R}^{n \times n}$ given by $B(Q)=\mathrm{exp}(-Q)\;A\;\mathrm{exp}(Q)$ is a cont. function of $Q$. It obviously is (with $B(0)=A$).
  2. No. As this answer shows, $\mathrm{exp}(Q)\in \mathrm{GL}^+_n(\mathbb{R})$ if $Q\in \mathbb{R}^{n\times n}.$
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Bobson Dugnutt
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