 Solution

Let $a=\mu (A)$, $b=\mu (B)$ and $c=\mu (A+B)$. The sum is also symmetric negative definite. Part (a) gives the exponential norm bounds, while the triangle inequality and submultiplicativity giveApplying these estimates to the exponential product defect identity yieldsIf $c<a+b$, evaluate the scalar integral to obtain the decaying bound for symmetric negative matrix splitting:The sign condition is natural: for every unit vector $v$, $v^T(A+B)v\le a+b$, so the Rayleigh quotient gives $c\le a+b$.

When $c=a+b$, the integrand equals $e^{tc}$ independently of $x$, givingThis also follows by taking the continuous limit of the quotient as $a+b-c\to 0$. There is no division-by-zero exception in the original integral argument. All exponents are negative, so these bounds decay for large $t$; they are coarse bounds and need not reproduce the sharper quadratic small-$t$ error.