Offdiagonale

Offdiagonale is a term used to describe elements that lie outside the main diagonal of a square matrix or grid. In mathematics, an entry a_ij is off-diagonal when i does not equal j; entries with i = j lie on the diagonal. The off-diagonal part comprises all positions not on the main diagonal.

In linear algebra, many problems simplify when the matrix is diagonal, meaning all off-diagonal elements are zero. Nonzero off-diagonal elements indicate couplings between different basis states or variables and carry information about interactions that are not captured by the diagonal alone.

In statistics, off-diagonal entries of a covariance or correlation matrix quantify relationships between distinct variables. The diagonal entries typically measure variances; nonzero off-diagonal entries reveal correlations. Zero off-diagonal entries indicate lack of linear correlation under suitable assumptions.

In quantum physics, off-diagonal elements of a density or state matrix reflect coherence between quantum states. Their presence signals superposition and interference. The concept of off-diagonal long-range order describes persistent coherence across a system, an important idea in superconductivity and Bose–Einstein condensates.

Notationally, for an n×n matrix, the off-diagonal set is {a_ij | i ≠ j}. The term offdiagonale is sometimes used in non-English texts to refer to this idea; in English, off-diagonal or non-diagonal are common. The distinction between diagonal and off-diagonal components helps analyze structure, stability, and physical properties.