By inspection, is is evident that the reflection matrix is an orthogonal matrix, while its transpose is equivalent to its non-transpose. The reflection matrix can be represented by the square matrix:
Now, geometrically it is evident that its transpose is equivalent to its non-transpose since the reflection matrix works also as its own inverse, and this can be viewed geometrically:
However, what I was wondering is whether there are any other matrices, perhaps of different dimensions, which depict the same nature, and whether there is a particular method for finding matrices of this nature.
A reflection is its own inverse, and since it's an isometry leaving the origin fixed, its matrix is an orthogonal matrix. The inverse of an orthogonal matrix is its transpose, but the inverse of this matrix must be itself. Theferore the transpose of this matrix is itself, i.e. it is symmetric.