Is there a name for this special matrix? $$C_{n\times n}={\begin{bmatrix}
    1 &amp; x_{12} &amp; \dots  &amp; x_{1n} \\
    {x_{12}}^{-1} &amp; 1 &amp; \dots  &amp; x_{2n} \\
    \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\
    {x_{1n}}^{-1} &amp; {x_{2n}}^{-1} &amp; \dots  &amp; 1
\end{bmatrix}}$$

Criteria

$$x_{ab} = \frac{1}{x_{ba}}, \qquad x_{ab}&gt;0, \qquad x_{aa} = 1$$

Alternatively

$$C \cdot C^{T} =  \underline1$$

I know that this matrix models currency exchange (without commission or fluctuation). So I'm guessing it's called a currency matrix or a trade matrix. I'm just after a name so I can search it's properties.

I couldn't spot it in the Matrix Reference Manual. http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/special.html

I'm curious about some of it's algebraic properties, for example it has only 1 non-zero eigenvalue. What does this eigenvalue signify?

$$\text{Eigenvalues of} \begin{bmatrix}
    1 &amp; 6 &amp; 30  &amp; 210 \\
    1/6 &amp; 1 &amp; 5  &amp; 35 \\
    1/30 &amp; 1/5 &amp; 1 &amp; 7 \\
    1/210 &amp; 1/35 &amp; 1/7  &amp; 1
\end{bmatrix} = 0,0,0,4$$

Any help, ideas, advice greatly welcomed. Thanks
$C$ is a positive reciprocal matrix

to be an exchange matrix the additional condition that $a_{ij} \times a_{jk} = a_{ik}$ is required

$C$ is then a Positive Saaty-consistent Reciprocal matrix (PS-cR)

the eigenvalues of an $n\times n$ PS-cR matrix are $n,0,..,0$

Thank you @Algebraic Pavel for the term reciprocal.
It is laplacian matrix which is normally I-A, where A is adjacency matrix.
https://en.m.wikipedia.org/wiki/Laplacian_matrix