Neural networks parameterize finite-rank generators for ODEs on the orthogonal Lie group, allowing optimization of orthonormal bases in function space with a universality result that rank-2 generators suffice for density.
Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible
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abstract
This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting -1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or -1. The second remark has to do with ways of flipping the signs of the entries of a diagonal matrix, C, with nonzero diagonal entries, obtaining a new matrix, E, so that E + A is invertible, where A is any given matrix (invertible or not).
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2026 1verdicts
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Learning Orthonormal Bases for Function Spaces
Neural networks parameterize finite-rank generators for ODEs on the orthogonal Lie group, allowing optimization of orthonormal bases in function space with a universality result that rank-2 generators suffice for density.