  The BMV conjecture states that for $n\times n$ Hermitian matrices $A$ and $B$
the function $f_{A,B}(t)=trace{\, } e^{tA+B}$ is exponentially convex. Recently
the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is
based on ingenious considerations related to Riemann surfaces of algebraic
functions. In the present paper we give a purely "matrix" proof of the BMV
conjecture for the special case $rank\,A=1$. This proof is based on the Lie
product formula for the exponential of the sum of two matrices and does not
require complex analysis.
