Matrices of SL${(4, \mathbb{R})}$ that are the Product of Two Skew-Symmetric Matrices

Abstract

The Jordan forms of matrices that are the product of two skew-symmetric matrices over a field of characteristic ${\neq 2}$ have been a research topic in linear algebra since the early twentieth century. For such a matrix, its Jordan form is not necessarily real, nor does the matrix similarity transformation change the matrix into the Jordan form. In 3-D oriented projective geometry, orientation-preserving projective transformations are matrices of ${SL(4, \mathbb{R})}$, and those matrices of ${SL(4, \mathbb{R})}$ that are the product of two skew-symmetric matrices are the generators of the group ${SL(4, \mathbb{R})}$. The canonical forms of orientation-preserving projective transformations under the group action of ${SL(4, \mathbb{R})}$-similarity transformations, called ${SL(4, \mathbb{R})}$-Jordan forms, are more useful in geometric applications than complex-valued Jordan forms. In this paper, we find all the ${SL(4, \mathbb{R})}$-Jordan forms of the matrices of ${SL(4, \mathbb{R})}$ that are the product of two skew-symmetric matrices, and divide them into six classes, so that each class has an unambiguous geometric interpretation in 3-D oriented projective geometry. We then consider the lifts of these transformations to SO(3, 3) by extending the action of ${SL(4, \mathbb{R})}$ from points to lines in space, so that in the vector space ${\mathbb{R}^{3, 3}}$ spanned by the Plücker coordinates of lines these projective transformations become special orthogonal transformations, and the six classes are lifted to six different rotations in 2-D planes of ${\mathbb{R}^{3, 3}}$.