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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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up vote 5 down vote accepted

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)

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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.

Some relevant terms here are the spectral theorem and Sylvester's law of inertia.

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