Sum of Triangular Matrices
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Theorem
Let $\mathbf A = \left[{a}\right]_{n}, \mathbf B = \left[{b}\right]_{n}$ be square matrices of order $n$.
Let $\mathbf C = \mathbf A + \mathbf B$ be the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.
If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, then so is $\mathbf C$.
If $\mathbf A$ and $\mathbf B$ are lower triangular matrices, then so is $\mathbf C$.
Proof
From the definition of matrix addition, we have:
- $\forall i, j \in \left[{1 .. n}\right]: c_{ij} = a_{ij} + b_{ij}$
If $\mathbf A$ and $\mathbf B$ are upper triangular matrices, we have:
- $\forall i > j: a_{ij} = b_{ij} = 0$
Hence:
- $\forall i > j: c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0$
and so $\mathbf C$ is itself upper triangular.
Similarly, if $\mathbf A$ and $\mathbf B$ are lower triangular matrices, we have:
- $\forall i < j: a_{ij} = b_{ij} = 0$
Hence:
- $\forall i < j: c_{ij} = a_{ij} + b_{ij} = 0 + 0 = 0$
and so $\mathbf C$ is itself lower triangular.
$\blacksquare$