# solution to a second order matrix equation

Let $$P_1 = \sum_{k=1}^m A_kA_k^T$$ and $$P_0 = \sum_{k=1}^m c_k(t)\cdot A_kA_k^T$$ with $$c_k(t) \in [a,b]$$ where $$a,b \in (0,1)$$. We will also assume that $$P_1$$ has full rank. Then it is obvious that $$P_0,P_1$$ are symmetric positive definite matrices.

## Question:

I want to find $$Q_1,Q_0$$ such that $$\begin{cases} Q_1 + Q_0 = P_1\\ Q_1\cdot Q_0 = P_0\end{cases}$$ Do these matrices, $$Q_1,Q_0$$ exist? If so, are their eigenvalues with positive real part?

## My attempt:

Since $$P_1,P_0$$ commute we will treat them as scalars ... Hence let $$Q_1 = P_1 - Q_0$$ then $$(P_1 - Q_0)\cdot Q_0 = P_0 \iff Q_0^2 - P_1\cdot Q_0 + P_0 = 0_{n\times 1}$$ From here follows $$Q_0 = \frac{1}{2} \cdot \left( P_1 + \left( P_1^2 - 4\cdot P_0\right)^{\frac{1}{2}}\right)$$

$$Q_1 = \frac{1}{2} \cdot \left( P_1 - \left( P_1^2 - 4\cdot P_0\right)^{\frac{1}{2}}\right)$$ If this is true, all that is left is to decide whether the eigenvalues of $$Q_1, Q_0$$ have positive real part or not ... Can some one confirm this and give some ideas regarding how to proceed for prooving the eigenvalue property?

$$P_0$$ and $$P_1$$ are diagonalizable and they commute. Therefore, they are simultaneously diagonalizable, i.e. there is an invertible matrix $$S$$ such that $$P_0 = SD_0S^{-1}$$ and $$P_1 = SD_1S^{-1}$$ with diagonal matrices $$D_0$$ and $$D_1.$$ Now we set $$Q_0=SC_0S^{-1}$$ and $$Q_1=SC_1S^{-1}$$ and we get $$\begin{eqnarray*} Q_1+Q_0 &=& SC_0S^{-1}+SC_1S^{-1} = S(C_0+C_1)S^{-1} \\ P_1 &=& \phantom{SC_0S^{-1}+SC_1S^{-1} = } SD_1S^{-1} \\ Q_1Q_0 &=& SC_0S^{-1}\cdot SC_1S^{-1} = SC_0C_1S^{-1} \\ P_0 &=& \phantom{SC_0S^{-1}\cdot SC_1S^{-1} = } SD_0S^{-1} \end{eqnarray*}$$ So it is obviously sufficient to solve $$C_0+C_1 = D_1 \\ C_0C_1 = D_0$$ for $$C_0$$ and $$C_1.$$ $$C_0$$ and $$C_1$$ in turn are diagonal matrices which can be found by looking at each diagonal element of $$D_0$$ and $$D_1$$ in isolation. It can now easily be verified that $$C_0$$ and $$C_1$$ are diagonal matrices with elements that have a positive real part.