I would like to cast some problems involving a symmetric hollow (diagonal entries equal to $0$) matrix in an equivalent form represented by the vectorization of the matrix. I know that if the matrix is only symmetric, then an effective way is to use the half-vectorization of the matrix and the use of duplication and elimination matrices. I would like to find something similar to the case of hollow symmetric matrices.
I have a real-valued function $f$ with argument a symmetric hollow matrix $\mathbf{A}$. Because all the information of the matrix is stored in the strictly lower (or upper) triangular part of the matrix, call it $\mathbf{A}_1$ , I guess there should be a way to rewrite the function by using the vectorization of the matrix $\mathbf{A}_1$, i.e. $\mathbf{a}=\mathbf{A}_1$.
For instance, considering a symmetric matrix $\mathbf{B}$:
$$ f(\mathbf{A})= \operatorname{trace}(\mathbf{A}\mathbf{B}) = 2 \mathbf{a}^{\top}\mathbf{b}$$
This is immediate with this particular function, but how can I generalize to other functions? For instance, what about $ f(\mathbf{A})= \operatorname{trace}(\mathbf{A}^\top\mathbf{A}\mathbf{B})$.
Can in general also the product $\mathbf{A}\mathbf{A}$ be rewritten with a particular reshape of the vectors product?
Thank you
EDIT
If I have function $f(\mathbf{A})$, can I compute in this case gradient and Hessian in the hollow-half-vec space, and being able to compute back the associated matrices? As a related subquestion, is the $\operatorname{vechh(\cdot)}$ an invertible operation?
Of course, once that I have $\mathbf{a}=\operatorname{vechh(\mathbf{A})}$ I can easily reshape the vector in a way to get back $\mathbf{A}$. However, how is this mathematically represented? I should be able to do something like:
$$ \mathbf{A}= \operatorname{vec}^{-1} (\mathbf{D}\mathbf{H}^{-1}\mathbf{a}) \mathbf{P}^\top= \operatorname{vec}^{-1} (\mathbf{D}\mathbf{H}^{-1}\mathbf{H}\mathbf{E}\operatorname{vec}(\mathbf{AP})) \mathbf{P}^\top $$
where $\mathbf{D}$,$ \mathbf{E}$ are the duplication and elimination matrices, respectively, and $\mathbf{H}^{-1}$ is the inverse of the "hollow" matrix $\mathbf{H}=[\mathbf{0}_{l\times n} \mathbf{I}_l]$.
As a case study, we could take $f(\mathbf{A})= \operatorname{trace}(\mathbf{A}\mathbf{B}) = 2 \mathbf{a}^{\top}\mathbf{b}$, with $\mathbf{A}$, $\mathbf{B}$ symmetric matrices, $\mathbf{A}$ hollow. It would be nice to have a methodology to express each function $f(\mathbf{A})$ in an equivalent form $f(\mathbf{a})$.
Best Answer
$\def\m#1{ \left[\begin{array}{r}#1\end{array}\right] }$ Given a hollow matrix $$\eqalign{ A &\in {\mathbb R}^{n\times n} \\ }$$ the hollow-half-vec operation is analogous to the more familiar half-vec operation. Both can be described in terms of the standard vec operator $$\eqalign{ {\rm vechh}(A) &= E_h\cdot{\rm vec}(A) \quad&\sim\quad {\rm vech}(A) &= E\cdot{\rm vec}(A) \\ {\rm vec}(A) &= D_h\cdot{\rm vechh}(A) \quad&\sim\quad \;\;{\rm vec}(A) &= D\cdot{\rm vech}(A) \\ }$$ where $E_h\in{\mathbb R}^{\ell\times n^2}$ is the hollow-elimination matrix. It is a sparse binary matrix $\,\ell = \tfrac 12(n^2-n)\,$. There is one non-zero element in each row, whose column index corresponds to the index of the conserved element in ${\rm vec}(A)$.
The hollow-duplication matrix is $D_h\in{\mathbb R}^{n^2\times\ell}$ is also a sparse binary matrix, whose elements are such that $E_hD_h=I$ and whose columns sum to two, i.e. $\;\tfrac 12D_h^T{\tt1} = {\tt1}.\;$ Again, this is analogous to the half-vec case where $ED=I\,$ (however the column sums of $D$ vary between one and two).
Interestingly, the pseudoinverse $D_h^+$ can serve as an elimination matrix, although $E_h^+$ fails as a duplication matrix, i.e. $$\eqalign{ {\rm vechh}(A) &= D_h^+\cdot{\rm vec}(A) \quad&\sim\quad {\rm vech}(A) &= D^+\cdot{\rm vec}(A) \\ {\rm vec}(A) &\ne E_h^+\cdot{\rm vechh}(A) \quad&\sim\quad \;\;{\rm vec}(A) &\ne E^+\cdot{\rm vech}(A) \\ D_h^+D_h &= I \quad&\sim\quad \quad D^+D &= I \\ }$$ $\Big[\,$In the hollow case, the calculation is particularly easy since $D_h^+ = \tfrac 12D_h^T\;\Big]$
For example, for $\;n=4,\;\ell=6$ $$\eqalign{ &E_h[1,2] &= 1,\qquad &D_h(2,1) &= 1,\quad &D_h(5,1) &= 1 \\ &E_h[2,3] &= 1,\qquad &D_h(3,2) &= 1,\quad &D_h(9,2) &= 1 \\ &E_h[3,4] &= 1,\qquad &D_h(4,3) &= 1,\quad &D_h(13,3)&= 1 \\ &E_h[4,7] &= 1,\qquad &D_h(7,4) &= 1,\quad &D_h(10,4)&= 1 \\ &E_h[5,8] &= 1,\qquad &D_h(8,5) &= 1,\quad &D_h(14,5)&= 1 \\ &E_h[6,12]&= 1,\qquad &D_h(12,6)&= 1,\quad &D_h(15,6)&= 1 \\ }$$ The indices of the vec and vechh vectors mapped onto $4\times 4$ matrices helps elucidate the components given above. $$\eqalign{ \m{1&5&9&13\\2&6&10&14\\3&7&11&15\\4&8&12&16} \qquad\qquad \m{0&1&2&3\\1&0&4&5\\2&4&0&6\\3&5&6&0} \\ }$$