**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](https://mathoverflow.net/q/347285/102946) 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?



Representation theorem for matrices (reference request)
My comments converted to an answer:



> 
> **1st comment.** I know no reference but the proof need not be [as per OP] lengthy — one just computes the two scalars by which the map $\mathscr I:$
> $$
> \textstyle A\mapsto\mathscr I(A)=\int\_S h\langle h,Ah\rangle\langle h,\cdot\rangle d\lambda(h)
> \tag1
> $$
>  acts on the irreducible components of $\mathfrak{gl}(n,\mathbf C)=\mathfrak{sl}(n,\mathbf C)\oplus\mathbf C I$ (Schur’s lemma).
> 
> 
> 


(In detail: write $r$ and $s$ for the scalars in question. Taking the trace of $sI=\mathscr I(I)$ in (1) gives $s=1$. It follows that we have
$
\mathscr I(A)= r\left(A-\tfrac{\operatorname{tr}A}{n}I\right)+\tfrac{\operatorname{tr}A}{n}I
$
and hence
$$
\operatorname{tr}(\mathscr I(A)B)=r\operatorname{tr}(AB)+\tfrac{1-r}n\operatorname{tr}(A)\operatorname{tr}(B)\rlap{\qquad\quad\forall A, B.}
\tag2
$$
Writing this out for $(A,B)=(E\_{12},E\_{21})$, resp. $(E\_{11},E\_{22})$ where $E\_{ij}=e\_i\langle e\_j,\cdot\rangle$, one obtains that $\int\_S|\langle e\_1,h\rangle|^2|\langle e\_2,h\rangle|^2d\lambda(h)$ equals both $r$ and $\frac{1-r}n$. Therefore $r=\frac1{n+1}$, and with that (2) becomes (3) below. QED)



> 
> **2nd comment.** For a *reference*: your desired formula is equivalent to
>  $$
> \mathrm{tr}(AB)+\mathrm{tr}(A)\mathrm{tr}(B)=(n+1)\int\_S\langle h,Ah\rangle\langle h,Bh\rangle\,d\lambda(h)
> \tag3
> $$
>  which is e.g. (3.8) of Gibbons ([1992](//ams.org/mathscinet-getitem?mr=93g:81031)), taking differences of notation into account. (The integrand descends to $P^{n-1}(\mathbf C)$ where he uses the measure of total volume $\pi^{n-1}/(n-1)!$) I think it should also follow from Archimedes-Duistermaat-Heckman ([1982](//ams.org/mathscinet-getitem?mr=84h:58051a), Prop. 3.2).
> 
> 
> 


More context: Generally I think you’ll find many similar formulas in the (somewhat repetitive) literature on “coherent states” or “quantum mechanics as classical mechanics on $P\mathscr H$”. Also compare Schur’s proof of his orthogonality relations ([1924](//zbmath.org/?q=an:50.0293.02), p.[199](//babel.hathitrust.org/cgi/pt?u=1&id=mdp.39015026454085&num=199) or Bröcker-tom Dieck [4.5i](//books.google.com/books?id=Kin7CAAAQBAJ&ots=srPKfNDHSN&pg=PA79)), specialized to the adjoint representation.


