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I know how to find the orthogonal and the special linear group of $2$ by $2$ matrices. This is because I know their “defining” properties. How can I find the Lie algebra of:

$$A = \left(\begin{array}[c c] - a_1 & a_2\\ 0& a_1^2 \end{array}\right),$$

Where the matrices are invertible? I don’t know their defining characteristic that is my big problem. If I know it, I can use dedicated to solve. Maybe it has to do something with their matrix potential or their trace?

Please help.

Bernard
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Dhdh
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    If you really mean that the matrix entry in the lower right is the square of the one top left, then this group is more specific than the "upper triangular" ones in the title; and the answer talks about those, not the ones with the extra condition in the question body. Please clarify. – Torsten Schoeneberg Jan 07 '20 at 01:48
  • I tried to put a general title. The group that I actually want is more specific and I denoted it by A. – Dhdh Jan 07 '20 at 02:07
  • Thanks, I get it now – Dhdh Jan 07 '20 at 13:10

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Let $f:Gl(2,\mathbb{R})\rightarrow \mathbb{R}$ defined by $f(\pmatrix{a&b\cr c&d})=c$, $f$ is a submersion, we deduce that $f^{-1}(0)$ the set of invertible upper triangular matrices $U(2,\mathbb{R})$ is a closed submanifold and a Lie subgroup of $Gl(2,\mathbb{R})$. This implies that the Lie algebra of $U(2,\mathbb{R})$ is the kernel of the differential of $f$ at the identity which is the subset of upper triangular matrices which are not necessarily invertible.

Tsemo Aristide
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