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A Cauchy matrix has entries $1/(x_i+y_j)$ with nonzero denominators. If $x_i=y_i>0$ are distinct, it is positive definite: $C_{ij}={\int }_0^{\infty }{e}^{-x_{i}s}{e}^{-x_{j}s}ds$ is the Gram matrix of linearly independent exponential functions. This form occurs in finite-rank inverse scattering transform reconstructions.

Choose distinct ${\kappa }_j>0$ and weights $c_j>0$ at spectral points $i{\kappa }_j$. With $E_j=e^{-{\kappa }_{j}x+{\kappa }_j^{3}t}$ and $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, solve $(I+A)\Phi =E$. The reconstructed multisoliton solution is $q=-{\partial }_x(c^T\Phi )$. Since $\Lambda A+A\Lambda =E{c}^T$, cyclicity of the trace gives $c^T\Phi =-2{\partial }_x\log \det (I+A)$. The Cauchy matrix $1/({\kappa }_i+{\kappa }_j)$ is positive definite, because it is the Gram matrix of $e^{-{\kappa }_{i}s}$ on $s>0$; diagonal similarity then proves $\det (I+A)>0$. For one spectral point this yields $q=({\kappa }^2/2){\mathrm{sech}}^2[\kappa (x-{\kappa }^{2}t-x_0)/2]$.

Take the spectral measure to have $n$ point masses at $k_j=i{\kappa }_j$, with distinct ${\kappa }_j>0$ and positive weights $c_j$. SetEvaluating the linear integral equation at these points gives the explicit algebraic system and reconstructionThe sign is fixed by $i/(i{\kappa }_i+i{\kappa }_j)=1/({\kappa }_i+{\kappa }_j)$. No continuum integration remains.

There is also a useful determinant form. Define $A_{ij}=c_j{E}_i/({\kappa }_i+{\kappa }_j)$, $B=I+A$, $\tau =\det B$ and $\Lambda =\mathrm{diag}({\kappa }_1,\dots ,{\kappa }_n)$. Then $A_x=-\Lambda A$ and $\Lambda A+A\Lambda =E{c}^T$. Using the derivative of a determinant and cyclicity of the trace,Hence the finite-rank KdV dressing determinant givesPositivity ensures a regular real solution: the Cauchy matrix $C_{ij}=1/({\kappa }_i+{\kappa }_j)$ is the Gram matrix of the independent functions $e^{-{\kappa }_{i}s}$ on $s>0$, so it is positive definite. The matrix $A=\mathrm{diag}(E)C\mathrm{diag}(c)$ is diagonally similar to $\mathrm{diag}(\sqrt{cE})C\mathrm{diag}(\sqrt{cE})$. Its eigenvalues are positive, and $\tau >0$. The resulting reflectionless multisoliton solution has $n$ distinct scales and velocities; the weights fix their positions. Arbitrary signed or complex weights need not give a nonsingular real solution.