offdiagonal

Offdiagonal, commonly written as off-diagonal or off-diagonal elements, refers to the components of a square matrix, tensor, or operator that lie outside the main diagonal. For a matrix A with indices i and j, the diagonal elements satisfy i = j, while the off-diagonal elements satisfy i ≠ j. The term is used across mathematics, physics, statistics, and related fields to distinguish cross-component interactions from self-interactions represented by diagonal terms.

In linear algebra, off-diagonal elements contribute to many matrix properties. A matrix is said to be diagonal if all off-diagonal elements are zero; in that case, many operations simplify. The pattern and magnitude of off-diagonal entries influence eigenvalues, conditioning, and the behavior of matrix functions. Symmetric matrices have A_ij = A_ji, so the same information appears twice in the off-diagonal portion. The off-diagonal norm or sparsity pattern is often used in numerical methods and in the construction of block-diagonal or sparse approximations.

In statistics and multivariate analysis, off-diagonal entries commonly describe relationships between different variables. In a covariance matrix, off-diagonal entries are covariances between distinct variables; in a correlation matrix, they are correlation coefficients. Large absolute values indicate strong linear relationships, though zero correlation does not necessarily imply independence except in specific distributions (e.g., multivariate normal). In precision (inverse covariance) matrices, sparsity of off-diagonal terms corresponds to conditional independence under Gaussian assumptions.

In physics and engineering, off-diagonal matrix elements frequently encode couplings or transitions between states, such as interaction terms in a Hamiltonian or Jacobian components that couple different variables. The concept also appears in differential equations, where off-diagonal terms describe how one equation or field influences another.

Overall, off-diagonal elements provide a measure of cross-component interaction and are central to analysis, modeling, and computation across disciplines.