The Transposition Operator and Notation



Let's say you have a matrix $A\in\mathrm{GL}_n$. How do we best denote the inverse of its transpose? You would probably write $(A^T)^{-1}$ or $(A^{-1})^T$ because it is the same. However, today at the office we decided to henceforth write $A^{-T}$ instead. It seems abusive at first, but I can make it formal for you, if you care for that kind of stuff. As we all know, the transpose of $A^T$ is $A$. In other words, the transposition operator $\vartheta:\mathrm{GL}_n\to\mathrm{GL}_n$ which maps $A$ to its transpose satisfies $\vartheta^2=1$. So far, the exponentiation map is defined as $\mathrm{GL}_n\times\mathbb{Z}\to\mathrm{GL}_n$ mapping $(A,k)\mapsto A^k$. We instead consider the ring $\mathbb{Z}[T]:=\mathbb{Z}[t]/(t^2)$ and extend the domain of the exponentiation map to $\mathrm{GL}_n\times\mathbb{Z}[T]_\ast$, where $\mathbb{Z}[T]_\ast$ denotes the homogeneous elements of $\mathbb{Z}[T]$. This way, you can write $A^{kT}$ instead of $(A^k)^T$ and you have $A^{kT^2}=A^k$ as required. Note that by restricting to homogeneous elements in $\mathbb{Z}[T]$, we get $A^{p+q}=A^pA^q$ and $A^{pq}=(A^p)^q$ for all $p,q\in\mathbb{Z}[T]_\ast$.

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