Dilations and characterisations of matrices

Let A, B be any two positive definite \(n\times n\) $n\times n$ matrices and Y be any \(n\times n\) $n\times n$ matrix. The matrices \(M_Y(A,B)=\left[ \begin{array}{cc} A &{} A^{\frac{1}{2}}YB^{\frac{1}{2}} \\ B^{\frac{1}{2}}Y^{\star }A^{\frac{1}{2}} &{} B \end{array}\right] \) $M_Y(A,B)=\left[\begin{array}{cc}A& A^{\frac{1}{2}}Y{B}^{\frac{1}{2}}\\ B^{\frac{1}{2}}{Y}^{\star }{A}^{\frac{1}{2}}& B\end{array}\right]$ for Y to be contractive, expansive or unitary matrix, are in fact arising from matrix/operator means. We aim to establish the signatures of the eigenvalues of the sum of two matrices of the type \(M_Y(A,B).\) $M_Y(A,B).$ We characterise any \(n\times n\) $n\times n$ matrix A through its dilations given by \({\mathcal {P}}_3(A)=\begin{bmatrix} O &{} A &{} A^2\\ A^* &{} O &{} A\\ {A^*}^2 &{} A^* &{} O \end{bmatrix}\) ${\mathcal{P}}_3(A)=\left[\begin{array}{ccc}O& A& A^2\\ A^{\ast }& O& A\\ {A^{\ast }}^2& A^{\ast }& O\end{array}\right]$ and \({\mathcal {M}}_3(A)=\begin{bmatrix} I &{} A &{} A^2\\ A^* &{} I &{} A\\ {A^*}^2 &{} A^* &{} I \end{bmatrix},\) ${\mathcal{M}}_3(A)=\left[\begin{array}{ccc}I& A& A^2\\ A^{\ast }& I& A\\ {A^{\ast }}^2& A^{\ast }& I\end{array}\right],$ by means of inertia of dilations.