# What is (are) the condition(s) for sum of a non-singular matrix and its transpose to be non-singular

Let a real (square) matrix $\mathbf A$ is Hurwitz (i.e., all the eigenvalues of $\mathbf A$ have negative real parts). And let $\mathbf P$ be a real symmetric positive definite matrix (i.e., $\mathbf {P = P^T > 0}$). Hence, both $\mathbf A$ and $\mathbf P$ along with their product $\mathbf {AP}$ are non-singular. What will be the condition(s) on $\mathbf A$ and/or $\mathbf P$ so that the sum $\mathbf {AP + PA^T}$ also becomes non-singular.

• If the product $\mathbf{AP}$ is symmetric then the summation $\mathbf{AP+PA^T}$ would become non-singular based on definiteness of both the matrices $\mathbf{A}$ and $\mathbf{P}$. But for generic non-symmetric matrix product $\mathbf{AP}$, what should be the condition for ensuring $\mathbf{AP+PA^T}$ to be non-singular. Can anyone please help in this regard? – neelarnab Aug 12 '14 at 18:17