offdiagonalnymi

Offdiagonalnymi is a Polish form derived from the English term off-diagonalny, used mainly in mathematical and scientific contexts to describe elements that lie outside the main diagonal of a matrix or similar structure. In standard Polish, more common expressions are poza przekątną or poza diagonalą; offdiagonalnymi represents the instrumental plural form of the borrowed adjective when describing multiple off-diagonal items.

In linear algebra, the off-diagonal elements of a square matrix A = [a_ij] are those with i ≠ j, while the diagonal elements satisfy i = j. The off-diagonal portion influences properties such as diagonalizability, symmetry, and sparsity. A matrix with only diagonal elements is diagonal; if all off-diagonal elements are zero, the matrix is diagonal. When off-diagonal elements exist, they can encode interactions, couplings, or cross-terms between different rows and columns.

Beyond pure linear algebra, off-diagonal elements appear in statistics, physics, and applied mathematics. In statistics, a covariance or correlation matrix has off-diagonal entries that quantify relationships between different variables. In physics, off-diagonal matrix elements can represent transitions, couplings, or interaction terms between states or modes, contributing to dynamics and spectral properties.

Terminology in Polish texts varies. You may encounter the exact anglicized form offdiagonalnymi, especially in technical writing with English borrowings, or more traditional phrases like elementy poza diagonalą. The meaning remains the same: referring to quantities not located on the matrix’s main diagonal.