Moore determinant - Knowlpedia

Definition

For an $n\times n$ hyperhermitian matrix $A$ , let $R_A$ be the real $4n\times4n$ matrix of the associated real-linear operator. The Moore determinant is the unique real homogeneous polynomial $\det_M A$ , normalized by $\det_M I=1$ , such that

\det_{\mathbb R}(R_A)=(\det_M A)^4.

It has degree $n$ and agrees with the ordinary determinant when $A$ has complex Hermitian entries.

Basic formulas

If $A=\operatorname{diag}(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\in\mathbb R$ , then $\det_M A=\prod_i\lambda_i$ . For

A=\begin{pmatrix}a&q\\ \bar q&b\end{pmatrix}, \qquad a,b\in\mathbb R,

one has $\det_M A=ab-|q|^2$ . In particular, the Moore determinant is nonnegative on positive-semidefinite hyperhermitian matrices.

Congruence law

For a quaternionic matrix $C$ ,

\det_M(C^*AC)=\det_M(A)\det_M(C^*C).

This is the replacement for multiplicativity needed in quaternionic Hessian geometry. The Moore determinant is not a determinant on all quaternionic matrices; it is defined on the hyperhermitian subspace. The Dieudonné determinant is a different construction with a different domain and codomain.

References

Semyon Alesker, “Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables,” Bulletin des Sciences Mathématiques 127 (2003), 1–35. arXiv record. Relevant: §§1.1–1.2.
Semyon Alesker, “Quaternionic plurisubharmonic functions and their applications to convexity,” 2016 revision. arXiv record. Relevant: §1.