I know that a matrix stands for some kind of linear transformation. such as
$$
\left(
\begin{matrix}
1&m\\
0&1
\end{matrix}
\right)
$$
as a shear mapping matrix. There are all kinds of transformations including rotation, reflection, scaling, shear mapping, squeeze mapping and projection.(Are there any more? Please list them out if you can.)
I try to apply some imagination to symmetric matrices, and I need more geometrical or visualizable interpretation, for this specific kind of matrix has so many useful properties.
But as for such a big category of matrices (symmetric matrices), I can't figure out a common interpretation or imagination. For example,
$$
\left(
\begin{matrix}
\frac{1}{2}&\frac{1}{2}\\
\frac{1}{2}&\frac{1}{2}\\
\end{matrix}
\right)
$$
is a symmetric matrix, and it's a projection matrix.
$$
\left(
\begin{matrix}
\frac{1}{2}&0\\
0&\frac{1}{2}\\
\end{matrix}
\right)
$$
is also a symmetric matrix, but it's a scaling one.
May be there are some more common and stronger interpretation(imagination/representation, anyway) for symmetric matrices, I don't know. May be you have some idea?
Best Answer
a real symmetric square matrix is orthogonally diagonalizable, that is to say, the linear transformation is just scaling in mutually perpendicular directions (perpendicular with respect to the basis you started with, but not necessarily parallel to your standard basis)