If we have an orthogonal matrix $U$, then $U^Tx$ is essentially a rotation of the vector $x$. If we have a diagonal matrix $\Lambda$, then $\Lambda x$ is scaling the vector $x$ in each direction by the corresponding diagonal value.
Since any symmetric matrix $S=U\Lambda U^T$, $Sx=U\Lambda U^Tx$ which is rotation of $x$ by some angle $\vartheta$, scaling it by $\Lambda$ and then re-rotating by angle $-\vartheta$. Does this imply that multiplying by any symmetric matrix $S$ is just a scaling by its eigenvalues since the net rotation is $0$?
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$\begingroup$Yes, but the orthogonal matrix $U$ determines in which directions this scaling happens. Only applying the diagonal matrix
$$\Lambda=\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$$
scales by two on the $x$-axis and by three on the $y$-axis.
But you can scale along other (orthogonal) axes if you want to by adding a rotation matrix.
You can imagine this by first transforming the coordinate system by $U^T$, so that the desired stretching axes line up with the coordinate axes (red and blue lines in the image), then you scale along the usual coordinate axes by $\Lambda$, and then you rotate back by $U$.
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