Union of two 3x3 matrices

I'm trying to find the Union of two sets of matrices. The first set is all diagonal matrices, for example $$A = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \\ \end{bmatrix} $$ and the second is the set of opposite diagonal matrices, for example $$B= \begin{bmatrix} 0 & 0 & c \\ 0 & b & 0 \\ a & 0 & 0 \\ \end{bmatrix} $$

I understand that the union is a collection of elements which are in A and B or both A and B, but i'm not sure how this works with matrices. Would it just be some set of matrices of the form $$C = \begin{bmatrix} a & 0 & b \\ 0 & c & 0 \\ d & 0 & e \\ \end{bmatrix} $$?

1 Answer

You have two sets of matrices: the set $\mathcal{A}$ of matrices of the form $A$ for $(a,b,c)\in\mathbb{R}$, and the set $\mathcal{B}$ of matrices of the form $B$.

I don't know what the union of matrices means, but I guess you are considering the union of the two sets. In that case, $C$ is not in $\mathcal{A}$ nor $\mathcal{B}$ for non-trivial values of $d$ and $c$. The union would then be the set of matrices of the form $A$ or $B$.

Note: the intersection would be the matrices of the form $$\begin{bmatrix} 0 & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & 0\end{bmatrix}$$.