Matrix Triangle Inequalities: Past, Present, and Future

We revisit several classical matrix inequalities related to the celebrated Thompson triangle inequality. We then present the second author's version of Thompson's inequality and related results. Finally, we survey a number of recent and elegant matrix triangle inequalities that may be viewed as cousins of Thompson's inequality. Taken together, these results illustrate the central role played by various notions of modulus in the noncommutative setting. Their interplay opens new perspectives in matrix analysis and operator theory.