Cauchy stress tensor
Definition 1
In physics, the square matrix whose components are stress and which is defined as follows $\mathbf{\sigma} \in \mathbb{R}^{3 \times 3}$ is called the Cauchy stress tensor. $$ \sigma = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} $$
Explanation
Although it denotes the stress acting at a point, because it includes shear stresses—forces that act tangentially to a surface— $\sigma$ is conveniently thought of, as shown in the definition, as an infinitesimal cube with a very small differential volume without losing generality.
Isotropy
In the context of fluid mechanics, a fluid is assumed to be isotropic, meaning the force exerted is the same in every direction. Isotropy is not a notion confined to fluid mechanics (see); it is an assumption encountered whenever directional dependence is neglected.
The diagonal components $\sigma_{11} , \sigma_{22} , \sigma_{33}$ are regarded as the normal stresses acting along the three coordinate directions, and if, under the assumption of isotropy, a uniform hydrostatic pressure $p$ is applied, then the trace of $\sigma$ satisfies the following. $$ \tr \left( \sigma \right) = \sigma_{11} + \sigma_{22} + \sigma_{33} = - 3 p $$
Here, the reason it is $-3p$ rather than $3p$ is that the force applied to the body is directed inward. If there are no stresses other than pressure, the Cauchy stress tensor can be simply written in terms of the identity matrix $I$ as follows. $$ \sigma = \begin{bmatrix} -p & 0 & 0 \\ 0 & -p & 0 \\ 0 & 0 & -p \end{bmatrix} = - p I $$