In Maths and Physics we often stumble across problems that can be described either by a triangular matrix or by a symmetric matrix. In particular it is usual to meet sums like the following: $$\sum_{j=1}\sum_{k=1}\mathcal{M}_{jk}x_{j}x_{k},$$where one can define a matrix $\mathcal{M}$ either as symmetric either as triangular (or others).
I guess this is no surprise as there is an isomorphism between symmetric and triangular matrices and they contain the same information. However I'm wondering about the advantages of each in terms of computations and the eventual relationships that there may or may not be between the two.
For example, suppose that in practical terms, I can write a matrix
$$\mathcal{M}_t=\begin{pmatrix}A & 0\\ 2B & C \end{pmatrix},$$where $A,B,C$ are also matrices and suppose I find its eigenvalues and eigenvectors. Is there something I can say about the eigenvalues and eigenvectors of $$\mathcal{M}_s=\begin{pmatrix}A & B\\ B & C \end{pmatrix}.$$ To further illustrate, suppose that $\mathbf{x}_{1}$ and $\mathbf{x}_{2}$ are column vectors:
\begin{align} \left(\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}\right)\underbrace{\begin{pmatrix}A & B\\ B & A \end{pmatrix}}_{\mathcal{M}_{s}}\begin{pmatrix}\mathbf{x}_{1}\\ \mathbf{x}_{2} \end{pmatrix} =\left(\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}\right)\begin{pmatrix}A\mathbf{x}_{1}+B\mathbf{x}_{2}\\ B\mathbf{x}_{1}+A\mathbf{x}_{2} \end{pmatrix} =\mathbf{x}_{1}^{T}A\mathbf{x}_{1}+\mathbf{x}_{1}^{T}B\mathbf{x}_{2}+\mathbf{x}_{2}^{T}B\mathbf{x}_{1}+\mathbf{x}_{2}^{T}A\mathbf{x}_{2},\end{align}On the other hand: \begin{align}\left(\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}\right)\underbrace{\begin{pmatrix}A & 0\\ 2B & A \end{pmatrix}}_{\mathcal{M}_{t}}\begin{pmatrix}\mathbf{x}_{1}\\ \mathbf{x}_{2} \end{pmatrix} =\left(\mathbf{x}_{1}^{T},\mathbf{x}_{2}^{T}\right)\begin{pmatrix}A\mathbf{x}_{1}\\ 2B\mathbf{x}_{1}+A\mathbf{x}_{2} \end{pmatrix} =\mathbf{x}_{1}^{T}A\mathbf{x}_{1}+2\mathbf{x}_{2}^{T}B\mathbf{x}_{1}+\mathbf{x}_{2}^{T}A\mathbf{x}_{2},\end{align} When $B$ is symmetric, these two expressions are equal. However, for $A=\begin{pmatrix}1 & 1\\ 1 & 1 \end{pmatrix}$ and $B=\begin{pmatrix}2 & 2\\ 2 & 2 \end{pmatrix}$ the spectra are totally different and the matrices do not even have the same definitness. So in summary, what is the relation between these two matrices? Is there anything well known about it? How are the eigenspectra related? And what would justify a particular choice of representation?