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

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 $B2$ ?

  • $\begingroup$ 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? $\endgroup$ – Leo Jul 31 at 20:04
  • $\begingroup$ I have edited the integral, but about the integrand of a matrix, as you mentioned it is entry-wise integrand @Leo $\endgroup$ – samie Aug 14 at 7:44

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