A private is a soldier, usually with the lowest rank in many armies. Soldiers with the rank of private may be conscripts or they may be professional (career) soldiers.
The term derives from the medieval term "private soldiers" (a term still used in the British Army), contrasting mercenary soldiers and denoting individuals who were either exclusively hired, conscripted, or mustered into service.
I am trying to learn tensor calculus with Bernacchi's book "Tensors made easy". For the inner product of two rank-2 tensors he gets four different results, each of them a rank-2 tensor. Why can't there be a fifth solution, a scalar? What is wrong with the following equation?
$$\mathbf A \cdot...