In $N > 2$ space there is a type of matrix that is unitary (which has ties to the hilbert space formulation of quantum mechanics). These matrices correspond to a constant rotation from any starting vector. Is there a concept of unitary matrices in a 1 dimensional line?
I know that a 1x1 matrix can be regarded as a scalar.
A 1×1 matrix is a number, and a unitary such is a phase, $e^{i\phi}$, which, in QM, may multiply wavefunctions without affecting the physics of expectation values.
Acting on infinite lines or circles, it represents the group of translations. You need to provide more context past this.