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I know that left multiplication of a vector $x$ by a orthonormal matrix $U$ is just changing the basis, but I'm not so sure what $U^TAU$ means. It seems that they have some relations.

Let $A$ be a symmetric matrix, and I take the symmetric matrix $A$ as a hyperellipsoid (in terms of $x^TAx=1$), $U^TAU$ seems to be just changing the basis to the direction of eigenvectors of $A$. In this way, $Ux$ is changing basis in terms of vector $x$ and $U^TAU$ is changing basis in terms of $A$.

If $B$ is any matrix viewed as a linear transformation, can I safely regard $U^TBU$ as changing the basis in terms of this linear transformation? How to verify this?

For example, will a horizontal shear mapping $C$ be change into a shear mapping in other direction if I use $U^TCU$? And will a scaling mapping in the direction of standard basis be change into a scaling mapping in other direction? If this is true, a symmetric matrix $A$ can be viewed as a scaling transformation in the directions of eigenvector of $A$. (https://math.stackexchange.com/a/103053/21762)

So, if $B$ is any matrix viewed as a linear transformation, can I safely regard $U^TBU$ as changing the basis in terms of this linear transformation?

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The reason is that for orthogonal matrices $U^T= U^{-1}$.

If you have a linear transformation $$x \mapsto Bx =y.$$ Then changing the basis (in the primed coordinates) with $U x'= x$ and $Uy'=y$, we have $$ Uy'= BUx' \Rightarrow y'= U^T B U x' = B' x'$$ such that $$B' = U^T B U$$ is the matrix of the linear transformation in the new coordinate system.

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