LAUSR

The Jordan forms of matrices that are the product of two skew-symmetric matrices over a field of characteristic $${\neq 2}$$≠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 $${SL(4, \mathbb{R})}$$SL(4,R), and those matrices of $${SL(4, \mathbb{R})}$$SL(4,R) that are the product of two skew-symmetric matrices are the generators of the group $${SL(4, \mathbb{R})}$$SL(4,R). The canonical forms of orientation-preserving projective transformations under the group action of $${SL(4, \mathbb{R})}$$SL(4,R)-similarity transformations, called $${SL(4, \mathbb{R})}$$SL(4,R)-Jordan forms, are more useful in geometric applications than complex-valued Jordan forms. In this paper, we find all the $${SL(4, \mathbb{R})}$$SL(4,R)-Jordan forms of the matrices of $${SL(4, \mathbb{R})}$$SL(4,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 $${SL(4, \mathbb{R})}$$SL(4,R) from points to lines in space, so that in the vector space $${\mathbb{R}^{3, 3}}$$R3,3 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 $${\mathbb{R}^{3, 3}}$$R3,3.