 Solution

Let $\beta =\mu [A]+\mu [B]$ and $\gamma =\mu [A+B]$, where all three matrices are symmetric. Submultiplicativity and the triangle inequality giveand similarly the other commutator term is bounded by $2\Vert B{\Vert }_2{e}^{x\beta }$. Apply part (a) also to $A+B$ in the integral from part (b). The outer factor one-half cancels these twos, leavingThe exponential divided difference evaluates this integral. Thuswhen $\beta \ne \gamma$, andwhen they coincide. The second expression is both the direct equal-exponent integral and the continuous limit of the first. The Rayleigh-Ritz variational principle also gives $\gamma \le \beta$, although the integral computation does not require a strict inequality. These are valid coarse norm bounds; the cancellation between the two products can make the actual small-step error substantially smaller.