Chapter 10.11, Problem 1MP

Show that the transformation equation for a $2^{\text{nd}}$-rank Cartesian tensor is equivalent to a similarity transformation.Warning hint: Note that the matrix $C$ in Chapter 3, Section 11, is the inverse of the matrix A we are using in Chapter 10 (compare $r^{\prime }=$ Ar and $r={\text{Cr}}^{\prime }$). Thus a similarity transformation of the matrix T with tensor components $T_{ij}$ is ${\text{T}}^{\prime }={\text{ATA}}^{-1}$.

Also see "Tensors and Matrices" in Section 3 and remember that $\text{A}$ is orthogonal.