Are the matrices which are subset of $GL_n(\mathbb{R})$,$\begin{pmatrix} 1&&a_{12}\\0&&a_{22}\end{pmatrix}$ and $\begin{pmatrix} a_{11}&&a\\0&&a\end{pmatrix}$ normal subgroups of $GL_2(\mathbb{R})$?
I tried finding the conjugate of any general invertible matrix with the above matrices but did not get an upper triangular matrix in the end. Any help. Thanks beforehand.
Best Answer
You definitely have subsets, but to be a normal subgroup you need for the sets to be subgroups which are closed under conjugation. Neither of them are since you can take 'most any matrix in $GL_2(\mathbb{R})$ with positive elements and conjugate 'most any matrix in your sets and get something which is not upper triangular, so definitely not in your sets.