**Motivation.** If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$ are the eigenvalues of $A$ (counting multiplicities), $(h_1,\dots,h_n)$ is a corresponding orthonormal basis of eigenvectors, and $h_k \otimes h_k$ denotes the matrix given by $(h_k \otimes h_k)x = \langle h_k, x\rangle h_k$ for each $x \in \mathbb{C}^n$ (here I used the "physical" convention that the inner product is linear in the second component).

**A representation result.** The following result for general (i.e. also non-normal) matrices is - very loosely - reminiscent of the above quoted spectral theorem:

Let $S$ denote the (Euclidean) unit sphere in $\mathbb{C}^n$ and let $\lambda$ denote the surface measure on $S$ (more precisely, we identify $\mathbb{C}^n$ with $\mathbb{R}^{2n}$, consider the surface measure on the unit sphere there and pull it back to $S$). Now, normalize $\lambda$ such that $\lambda(S) = n$.

**Theorem.** For every matrix $A \in \mathbb{C}^{n\times n}$ we have
$$
A = (n+1) \int_{S} \langle h, Ah \rangle \; h \otimes h \; d \lambda(h) - \operatorname{tr}(A) \, I;
$$
here, $\operatorname{tr}(A)$ denotes the trace of $A$ and $I \in \mathbb{C}^{n\times n}$ denotes the identity matrix.

One can prove the above theorem by using the answers to this MathOverflow question.

**The question (a reference request).** I have no idea whether the above representation theorem is of any use - but given its very symmetric and rather simple nature, it is natural to suspect that the theorem should already be somewhere in the literature.

So my question is: Do you know any reference where the above representation theorem is stated and proved?