In the recent paper by Tao concerning the famous eigenvalue-eigenvector identity, I need some insight regarding the "Dilation Symmetry" from a physical perspective.

It is mentioned that if we assign units to the entries of $A$, then the eigenvalues of $A$, $M_j$ acquire the same units, while $v_{i,j}$ remains dimensionless, and the identity (2) is dimensionally consistent.

The actual identity (2) given in the paper is :
(Eigenvector-eigenvalue identity). $$ \left|v_{i, j}\right|^{2} \prod_{k=1 ; k \neq i}^{n}\left(\lambda_{i}(A)-\lambda_{k}(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_{i}(A)-\lambda_{k}\left(M_{j}\right)\right) \tag{2} $$ where $A$ is an $n \times n$ Hermitian matrix, and its $n$ real eigenvalues are $\lambda_{1}(A), \ldots, \lambda_{n}(A)$, $M_{j}$ denote the $n-1 \times n-1$ minor formed from $A$ by deleting the $j^{\text {th }}$ row and column from $A$ and $v_{i, j}$ denote the $j^{\text {th }}$ component of $v_{i}$.

And Dilation Symmetry is defined as "If one multiplies the matrix $A$ by a real scalar $c$, then the eigenvalues of $A$ and $M_j$ also get multiplied by c, while the coefficients $v_{i,j}$ remain unchanged, which does not affect the equation (2)

It would be very helpful of someone can actually provide an physical example of this "Dilation Symmetry". Because personally I couldn't understand what kind of If you could provide me with some physical example of this "Dilation Symmetry", that would be extremely useful.


1 Answer 1


The moment of inertia tensor could be an example. https://en.wikipedia.org/wiki/List_of_moments_of_inertia#List_of_3D_inertia_tensors

If one doubles the mass of a solid ellipsoid,
then the eigenvalues double, but the principal directions remain the same.


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