I want to compute the determinant of the Hessian
$$ \boldsymbol H = \begin{bmatrix} \boldsymbol A & \boldsymbol B & \boldsymbol C \\ \boldsymbol D & \boldsymbol E & \boldsymbol F \\ \boldsymbol G & \boldsymbol H & I \end{bmatrix} $$
to check for positive definiteness.
Here, $\boldsymbol A, \boldsymbol B, \boldsymbol D, \boldsymbol E$ are fourth order tensors in a three-dimensional space, whose coefficients can be written as a $9\times9$ matrix. $\boldsymbol C, \boldsymbol F, \boldsymbol G, \boldsymbol H$ are second order tensors, whose coefficients can be written as a $1\times9$ matrix. $I$ is a scalar. All coefficients are real numbers. That said, $\boldsymbol H$ represents a $19\times19$ matrix. How can I exploit the block structure to compute the determinant? Are there special treatments if the blocks are rank2 / rank4 tensors?