Geometric interpretation of a hollow symmetrical 3D matrix Any matrix $A$ can be presented as a sum of its symmetrical and skew-symmetrical part:   
$A=sym(A)+skew(A)$.   
Decomposition can go further and we can present symmetrical part as a sum of some diagonal matrix and hollow (zeros on diagonal) symmetrical matrix. Now we have (used below notation is only for the needs of this question)
$A= diag(A)+hols(A)+ skew(A)$.
Let's assume the dimension of $A$ is $3$.   
In this case we have the sum of   $3$ components where every component has $3$ DOF (decoded in three 3D vectors). We can also present these components in the form
$A= k_d{diag_n}(A )+k_h{hols_n}(A )+ k_s{skew_n}(A )$, 
where coefficients 
$k_d, k_h, k_s$ 
are  calculated so to assure that vectors which represent DOF of components are unit vectors, index $n$ denotes here this kind of "normalization"
 (so in these unit vectors we have decoded 2 degrees of freedom and additional DOF is in the appropriate scaling coefficient $k_{\{d,h,s)\}}$)
Example of such decomposition:
$\begin{bmatrix}  
1 & 3 & 5 \\
1 & 4 & 4 \\
1 & 8 & 8
\end{bmatrix}$ = 
9$\begin{bmatrix}  
\dfrac{1}{9} & 0 & 0 \\
0 & \dfrac{4}{9} & 0 \\
0 &0 & \dfrac{8}{9}
\end{bmatrix}$+
7$\begin{bmatrix}  
0 & \dfrac{2}{7} & \dfrac{3}{7} \\
\dfrac{2}{7} & 0 & \dfrac{6}{7} \\
\dfrac{3}{7} & \dfrac{6}{7}  & 0 
\end{bmatrix}$+
3$\begin{bmatrix}  
0 & \dfrac{1}{3} & \dfrac{2}{3} \\
-\dfrac{1}{3} & 0 & -\dfrac{2}{3} \\
-\dfrac{2}{3} & \dfrac{2}{3} & 0
\end{bmatrix}$
Two of components have distinct geometric interpretation:
$diag(A)$ is simply a scaling matrix and $skew(A)$ is a scaled composition of a projection and  rotation by $\pi/2$ (See question*).
However the geometric interpretation for the $hols_n(A)$ is unknown to me..
In the general case matrix  $hols(A)$ has a form   
$\begin{bmatrix}  
0 & a & b \\
a & 0 & c \\
b & c & 0
\end{bmatrix}$
Some properties of this matrix    maybe  provide some information about its nature:


*

*it has real eigenvalues (as a special case of symmetric matrix) so
rotation probably can't be engaged for this interpretation (unless it
is by $\pi$)

*determinant of this matrix  $det(hols(A))= 2abc$ so its rank $=3$      if $ a,b,c \neq 0$. If one of $a,b,c=$ $0$ its rank decreases
(unlike $skew_n(A)$ where zeroing a single entry (and its skew-symmetrical one) doesn't change geometric interpretation - only axis of rotation which determines at the same time the direction of projection).

*few others  are  in Wikipedia.
So my question is:

  
*
  
*Could   hollow symmetrical matrix $hols_n(A)$ be decomposed in such a way that geometric    interpretation of $hols_n(A)$ would be explicit ?
  

If geometric interpretation of the hollow matrix is too difficult one can propose other method for decomposition of symmetrical part $sym(A)$ only on condition that both parts should have 3 DOF and each part should have explicit geometric interpretation..
 A: Just as a note aside, if instead of making it hollow, you fill the diagonal with $f_{1,\,2,\,3} (a,b,c)$
then you can get some significant factorization. For example:
$$
\left( {\begin{array}{*{20}c}
   1 & a & b  \\
   a & {1 + a^2 } & c  \\
   b & c & {1 + b^2  + (c - ab)^2 }  \\
 \end{array} } \right) = \left( {\begin{array}{*{20}c}
   1 & 0 & 0  \\
   a & 1 & 0  \\
   b & {c - ab} & 1  \\
 \end{array} } \right)\left( {\begin{array}{*{20}c}
   1 & a & b  \\
   0 & 1 & {c - ab}  \\
   0 & 0 & 1  \\
 \end{array} } \right)
$$
Now, doing that, the 3 DOF are preserved ; but, taking them out of the diagonal matrix,  the DOF there get increased.
I presume it is related to @user1551's note, but there is much to dig about..
A: What you are proposing is and additive transformation of coordinates.
$$
\mathbf{A} = diag\left( \mathbf{A} \right) + hollow\left( \mathbf{A} \right) 
$$
corresponds to that
the new vector $\mathbf{x}' = \left( {x'_1 ,x'_2 ,x'_3 } \right)$ 
is built first by $\left( {x'_1 ,0 ,0 } \right)$ (which is parallel to $\mathbf{x}$ )
plus $ \left( {0 ,x'_2 ,x'_3 } \right)$  (which is normal to $\mathbf{x}$ and to the previous step). Same for  $\mathbf{y}'$ and  $\mathbf{z}'$.
So a general Hollow matrix brings each base vector onto the plane normal to it. That is a rotation of $\pi/2$ around an axis normal to $\mathbf{x}$ ($1$ DOF) and a stretch ($1$ DOF): tot. $6$ DOF.
Symmetric and Skew-symmetric components do the same projections of each axis onto a plane normal to it, but the DOFs split into $3$ per each component, due to interrelation among rotation axes and stretch factors.
Of course the interrelation is different between the two components, but in this interpretation I cannot see a clearcut among the two.
