I was reading lecture notes which mentioned the set of upper (and separately lower)-triangular matrices of a certain dimensionality is a group under matrix multiplication. That made me wonder if they also form a ring under addition and multiplication.
So first, they are an abelian group under matrix addition:
- The sum of any number of triangular matrices is itself a triangular matrix.
- The 0 matrix is the 0 element.
- There is an additive inverse. (Element-wise negation)
- Matrix addition is commutative.
Then, they are a monoid under multiplication.
- The product of any number of triangular matrices is itself a triangular matrix.
- The identity matrix is the multiplicative identity.
And finally, multiplication distributes over addition.
Is that correct?