The equation X^⊤AX=B with B skew-symmetric: how much of a bilinear form is skew-symmetric?

ABSTRACT

Given a bilinear form on ${\mathbb{C}}^n$, represented by a matrix $A\in {\mathbb{C}}^{n\times n}$, the problem of finding the largest dimension of a subspace of ${\mathbb{C}}^n$ such that the restriction of A to this subspace is a non-degenerate skew-symmetric bilinear form is equivalent to finding the size of the largest invertible skew-symmetric matrix B such that the equation $X^{\mathrm{\top }}AX=B$ is consistent (here $X^{\mathrm{\top }}$ denotes the transpose of the matrix X). In this paper, we provide a characterization, by means of a necessary and sufficient condition, for the matrix equation $X^{\mathrm{\top }}AX=B$ to be consistent when B is a skew-symmetric matrix. This condition is valid for most matrices $A\in {\mathbb{C}}^{n\times n}$. To be precise, the condition depends on the canonical form for congruence (CFC) of the matrix A, which is a direct sum of blocks of three types. The condition is valid for all matrices A except those whose CFC contains blocks, of one of the types, with size smaller than 3. However, we show that the condition is necessary for all matrices A.

KEYWORDS:

• Matrix equation
• consistency
• transpose
• congruence
• canonical form for congruence
• skew-symmetric matrix
• bilinear form

AMS SUBJECT CLASSIFICATIONS:

• 15A21
• 15A24
• 15A63

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research has been funded by the Agencia Estatal de Investigación of Spain through grants MTM2017-90682-REDT, AEI/10.13039/501100011033.