10. Tensor Invariants

10.1. Rank-\(2\) tensors

• Characteristic polynomial (this definition needs that determinant exists)

  \[0 = \text{det}\left( \mathbf{A} - s \mathbf{I} \right) \ .\]

  Using a unit normal basis (todo generalization required?)

  \[\begin{split}\begin{aligned} 0 & = \left|\begin{matrix} A_{11} - s & A_{12} & A_{13} \\ A_{21} & A_{22} - s & A_{23} \\ A_{31} & A_{32} & A_{33} - s \end{matrix}\right| = - s^3 + s^2 I_1 - s I_2 + I_3 \ , \end{aligned}\end{split}\]

  with

  \[\begin{split}\begin{aligned} I_1 & = \text{tr}(\mathbf{A}) = A_{11} + A_{22} + A_{33} \\ I_2 & = \dfrac{1}{2} \left( \text{tr}(\mathbf{A})^2 - \text{tr}(\mathbf{A}^2) \right) = \dots = A_{11} A_{22} + A_{11} A_{33} + A_{22} A_{33} - A_{31} A_{13} - A_{12} A_{21} - A_{32} A_{23} \\ I_3 & = \text{det}(\mathbf{A}) = \dots \\ \end{aligned}\end{split}\]

  The coefficients of the characteristic polynomial are invariant under transformation of coordinates (todo here only referring to Cartesian basis, and thus orthogonal transformations?)

• Trace…

• Determinant…