Discovery.Linear­Disentanglement.Collinear­Ambiguity

For a $d$-dimensional coordinate system, two selected coordinates, and three real numbers specifying a line normal and deformation magnitude, the elementary two-row shear is the identity matrix plus $tv$ at position $(i,j)$ and $tu$ at position $(j,i)$. For distinct coordinates these are two off-diagonal entries; if the coordinates coincide, both increments land on the same diagonal entry.

For a $d\u2011by\u2011$d reference matrix, two selected coordinates, and a shear coefficient and magnitude, the first normalization denominator is $1+t v B_{ji}$.

For a $d\u2011by\u2011$d reference matrix, two selected coordinates, and a shear coefficient and magnitude, the second normalization denominator is $1+t u B_{ij}$.

For a $d\u2011by\u2011$d reference matrix, two selected coordinates, and a line normal and deformation magnitude, the pair row normalizer is the diagonal matrix that rescales the first selected row by the reciprocal of the first normalization denominator, the second selected row (when it differs from the first) by the reciprocal of the second, and leaves all other rows unchanged; a zero denominator has reciprocal zero by convention.

For a $d\u2011by\u2011$d reference matrix, two selected coordinates, and a line normal and deformation magnitude, the normalized two-coordinate deformation is the pair row normalizer multiplied by the elementary pair shear.

For a $d\u2011by\u2011$d reference diagonalizer, two selected coordinates, and a line normal and deformation magnitude, the deformed diagonalizer is the normalized two-coordinate deformation applied to the rows of the reference diagonalizer.

For a $d\u2011by\u2011$d matrix and two selected coordinates, pair-cycle admissibility holds exactly when the product of the $(i,j)$ and $(j,i)$ entries is not one; for distinct coordinates these are the two opposite off-diagonal entries of the selected pair.

For a $d\u2011by\u2011$d reference matrix, two selected coordinates, and a line normal, offset, and deformation magnitude, the common shift cross term is the line offset times the deformation magnitude, divided by both selected normalization denominators.

For a $d$-dimensional coordinate system, two selected coordinates, and a real entry value, the symmetric selected off-diagonal matrix is the sum of the matrix with that value at position $(i,j)$ and the matrix with that value at position $(j,i)$. For distinct coordinates it has the value in both off-diagonal positions and zero elsewhere; if the coordinates coincide, it has twice the value on that diagonal entry.

For a $d\u2011by\u2011$d reference diagonalizer and invariant matrix, two selected coordinates, and a line normal, offset, and deformation magnitude, the deformed invariant matrix is obtained by first forming the normalized two-coordinate deformation, then taking its congruence transform of the invariant matrix and adding the symmetric common shift cross term.

For a $d\u2011by\u2011$d reference diagonalizer, two selected coordinates, a line normal and deformation magnitude, an original diagonal shift vector, and a coordinate, the deformed shift at that coordinate is the corresponding diagonal entry obtained after forming the normalized two-coordinate deformation and taking the congruence transform of the original diagonal shift matrix.

For a $d\u2011by\u2011$d diagonalizer and invariant matrix and a diagonal shift vector, the represented covariance matrix is the inverse congruence transform of the invariant matrix plus the diagonal shift matrix. This is meaningful for an invertible diagonalizer; for a singular one the matrix inverse is zero by convention, so the value is the zero matrix.

For a dimension, an environment collection, a covariance matrix for each environment, a diagonalizer and invariant matrix, and a diagonal shift vector for each environment, the covariance family is represented exactly when every environment's covariance matrix equals the covariance represented by those common matrices and that environment's shift vector. Invertibility of the diagonalizer is not part of this predicate; a singular diagonalizer represents only the all-zero family.