Hyperhermitian form - Knowlpedia

Definition

Let $V$ be a finite-dimensional right quaternionic vector space. A hyperhermitian form is a map $a:V\times V\to\mathbb H$ that is additive in each variable and satisfies

a(x,yq)=a(x,y)q, \qquad a(x,y)=\overline{a(y,x)}.

Thus it is right-linear in the second variable and conjugate-linear in the first. It is positive definite if $a(x,x)>0$ for every $x\ne0$ .

Matrix presentation

After choosing a basis, $a$ is represented by a matrix $A=(a_{ij})$ with $A^*=A$ , equivalently $a_{ij}=\overline{a_{ji}}$ , through

a(x,y)=\sum_{i,j}\overline{x_i}a_{ij}y_j.

The representing matrix is therefore self-adjoint over $\mathbb H$ . Under a change of basis by $C$ , it transforms by congruence as $A\mapsto C^*AC$ .

Spectral theorem and positivity

Every hyperhermitian matrix is unitarily diagonalizable with real diagonal entries. It is positive semidefinite exactly when those entries are nonnegative. The Moore determinant is the determinant adapted to this self-adjoint class.

Terminology

“Hyperhermitian” here describes linear algebra over $\mathbb H$ . A hyper-Hermitian manifold is a geometric structure whose tangent-space metrics give such forms, but it also includes a smoothly varying hypercomplex triple.

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.
Fuzhen Zhang, “Quaternions and matrices of quaternions,” Linear Algebra and its Applications 251 (1997), 21–57. DOI record00543-9).