# what property between two matrices affects on the trace of multiply of them

I have two matrices $$A$$ with dimension $$n\times n$$ and $$B$$ with dimension $$n\times m$$ that $$m.

In my problem matrix, $$A$$ is fixed and given, but the elements of the $$B$$ are variables that can be continuous between $$0$$ and $$1$$ or discrete binary $$0$$ or $$1$$, but it's not important now.

I am working on an equation based on the $$A$$ and $$B$$ that is in the blow:

$$x =\int_0^1 e^{At}BB^{T}e^{A^{T}t}{dt}$$

here $$t$$ is a constant equal to $$1$$ and my goal is the $$\text{Trace}(x^{-1})$$.

I want to ask that, is there any property or relation between given matrix $$A$$ and arbitrary matrix $$B$$ that by that property or relation I will be able to say that matrix $$B1$$ will have greater $$\text{Trace}(x^{-1})$$ than matrix $$B2$$ ?

• The equation is a bit confusing since the upper limit of the integral is denoted by the same symbol as the variable in the integrand ($t$). Also, the integrand is a matrix, what does the integral of a matrix mean? Is it entry-wise? – Leo Jul 31 at 20:04
• I have edited the integral, but about the integrand of a matrix, as you mentioned it is entry-wise integrand @Leo – samie Aug 14 at 7:44