# Rotation-invariant matrix operations, like Frobenius inner product?

Rotation (change of basis) is a natural matrix transformation (let's focus on real $\mathbb{R}^{n\times n}$ here): $$r_O(A)=O^TAO\qquad \textrm{for some orthogonal}\qquad O^TO=OO^T=1$$

There are well known 1-matrix operations which are rotation invariant - trace of powers and characteristic polynomial: $$\textrm{Tr}(A^k)=\textrm{Tr}(r_O(A)^k)\qquad \qquad \det(A-\lambda I)=\det(r_O(A)-\lambda I)$$ generating two families of symmetric polynomials of eigenvalues.

I have just realized that there is natural 2-matrix rotation invariant Frobenius inner product ("scalar product for matrices"): $$\langle A, B\rangle_F:=\textrm{Tr}(AB^T)=\sum_{ij} A_{ij} B_{ij}\qquad \textrm{inducing}\qquad \|A\|_F^2=\sum_{ij}A_{ij}^2=\textrm{Tr}(AA^T)$$ Frobenius norm. Its rotation-invariance is easy to check: $\langle r_O(A),r_O(B)\rangle_F=\langle A, B\rangle_F.$

What other rotation-invariant operations on matrices are known?

I am mostly interested in real symmetric matrices - my motivation is that graph isomorphism problem can be transformed into testing if $\mathcal{A}=\{a_1 A_1+\ldots+a_m A_m\}$ and $\mathcal{B}=\{b_1 B_1+\ldots+b_m B_m\}$ linear spaces of symmetric matrices differ only by rotation (stack) - the question is how to test it effectively?

Update 1: Analogously to Frobenius inner product, $\det(AB^T-\lambda I)$ is also rotation-invariant, hence the entire eigenspectrum of $AB^T$ is rotation-invariant. Analogously for larger number of matrices: $\det(ABC-\lambda I)=\det(r_O(A)r_O(B)r_O(C)-\lambda I)$. Are there also essentially different constructions?

Update 2: Regarding testing if two linear spaces ($\mathcal{A}$ and $\mathcal{B}$) of symmetric matrices differ only by rotation, I think I have a way. Using $\textrm{Tr}(A)$ is dangerous as it is often 0 (e.g. for testing graph isomorphism). $1=\textrm{Tr}(A^2)=\sum_{ij} a_i a_j \textrm{Tr}(A_i A_j)$ defines elipsoid in $\mathcal{A}$, which has to correspond to analogous ellipsoid in $\mathcal{B}$ - we can use it to rescale to sphere. Then $\textrm{Tr}(A^3)$ should allow to uniquely define $m-1$ points on such sphere: first as the one maximizing $\textrm{Tr}(A^3)$ (unique?), second as maximizing it in orthogonal direction, and so on - then it would be sufficient to test if these points agree for $\mathcal{A}$ and $\mathcal{B}$.

Update 2.1: Unfortunately $\sum_i x_i^3$ has exponential number of maxima on unit sphere, so maximizing $\textrm{Tr}(A^3)$ won't work. A different than $\textrm{Tr}(A^2)$ quadratic, also inexpensive to calculate, is $\det(A-\lambda I)[\lambda^{n-2}]=\pm \sum_{i<j} \lambda_i \lambda_j$, giving some hope here. Other are generalized characteristic polynomials.

Update 3: $\det(A-\lambda I)[\lambda^{n-2}]=\pm \sum_{i<j} \lambda_i \lambda_j$ has turned out identical as $\textrm{Tr}(A^2)$. However, $\textrm{Tr}(A^3)$ as homogeneous polynomial, can be effectively describe with rotation invariants.

• Bhatia's Matrix Analysis has an interesting result on unitarily invariant norms: in particular, every such norm can be described as a function applied to the singular values of the matrix – Omnomnomnom Dec 15 '17 at 14:34
• You may also look at math.stackexchange.com/questions/1782042/… – Widawensen Dec 15 '17 at 16:14