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?

  • 4
    $\begingroup$ Your reasoning looks correct to me. $\endgroup$ – M. Winter Jun 26 '18 at 8:09
  • $\begingroup$ All requirements for definition en.wikipedia.org/wiki/Ring_(mathematics)#Definition seem to be satisfied..See also en.wikipedia.org/wiki/Matrix_ring $\endgroup$ – Widawensen Jun 26 '18 at 9:54
  • $\begingroup$ You probably mean the set of upper triangular matrices, not the set of upper or lower triangular matrices. $\endgroup$ – lhf Jun 26 '18 at 13:02
  • $\begingroup$ If: Yes. That is what I was trying to get to with "unitriangular" matrices. The set of upper-triangular matrices of a particular dimensionality and separately, the set of lower-triangular matrices of a certain dimensionality. $\endgroup$ – azani Jun 26 '18 at 15:50

Yes. The reasoning above is correct.


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