I was looking at the proof of the stability of a problem via the Lyapunov function in a paper. I don't think it is necessary to write the whole problem fully so I will focus just on the particular issue. Somewhere in the proof, they wrote that: $$ \Theta^T*(\Psi^{T}\Gamma) *\Theta = \Theta^T*(\Gamma\Psi) *\Theta $$
Where: $\Theta$ is a $p \times n$ matrix of the parameter errors of the system (real matrix, time varying). $\Psi$ is a $p \times p$ matrix (real matrix, not time-varying). $\Gamma$ is a $p \times p$ symmetric positive definite matrix (real diagonal matrix, not time-varying). The paper earlier defined a relationship between $\Psi$ and $\Gamma$ as: $$ \Gamma * \Psi + \Psi^T*\Gamma > 0 $$ Which in my understanding means simply that $(\Gamma * \Psi)$ is positive definite. So my question is how can we assert the first equation based only on the fact that $(\Gamma * \Psi)$ is positive definite? It seems to me, we must also assert that $(\Gamma * \Psi)$ is symmetric to be able to conclude that, no? The paper did not mention anything about $\Psi$ except it exists.
I appreciate the help. Thanks. dd
P.S: Now that I think of it, perhaps the authors meant to say that: $$ Trace[\Theta^T*(\Psi^{T}\Gamma) *\Theta] = Trace[\Theta^T*(\Gamma\Psi) *\Theta] $$ (which is obviously true if the 1st equation is correct, but the context of the problem was stability proof, so maybe it was a typo?) Still, it's not quite obvious to me how the two Trace expressions are equal. NB: in my question, "*" represents ordinary everyday matrix multiplication. I apologise if any confusion, I'm a new user here and still trying to get used to the commands for the proper math symbols. Thanks.