# What is the transformation representation/interpretation of symmetric matrices?

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?

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I think things are clearer if, instead of thinking of symmetric matrices in terms of linear transformations, you think of them in terms of symmetric bilinear forms $x^T M y$, or equivalently (assuming you're working over $\mathbb{R}$), in terms of quadratic forms $x^T M x$. If $M$ is positive-definite, then these quadratic forms can in turn be understood in terms of their "unit spheres" $x^T M x = 1$, which are ellipsoids. In general the unit spheres can be more complicated shapes like hyperboloids or ellipsoids / hyperboloids in a lower-dimensional subspace.