[Math] Determinant of sum of Kronecker products

determinantslinear algebranumerical linear algebra

Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant:

$\det|A \otimes C + B \otimes D |$

Best Answer

The case $A$ symmetric positive definite and $B$ symmetric and $C$ positive definite and $D$ symmetric is slightly easier.

In that case, first write $P'AP=I$ and $P'BP=\Delta_1$, and $Q'CQ=I$ and $Q'DQ=\Delta_2$, where $\Delta_1$ and $\Delta_2$ are diagonal matrices.

Now, write $(P\otimes Q)'(A\otimes C)(P\otimes Q) + (P\otimes Q)'(B\otimes D)(P\otimes Q)$ equals $(P'AP \otimes Q'CQ) + (P'BP \otimes Q'DQ) = (I\otimes I) + (\Delta_1\otimes \Delta_2)$.

Note that it takes $O(n^3+m^3)$ time to compute $P$ and $Q$.

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