Diagonalelementer

Diagonalelementer are the elements of a matrix that lie on its main diagonal, that is, the entries aij for which the row index i equals the column index j. In an m-by-n matrix A, there are min(m, n) diagonal elements. For a square matrix of size n, the diagonal elements are a11, a22, …, ann.

The diagonal part of a matrix is often denoted diag(A) and refers to a diagonal matrix that has the diagonal elements of A on its main diagonal and zeros elsewhere. The concept is central in many linear-algebraic operations, including the computation of the trace, which is the sum of the diagonal elements: tr(A) = a11 + a22 + … + ann for a square matrix.

Examples and special cases help illustrate the idea. Consider A = [[4, 1, 0], [0, 3, 2], [5, 0, 6]]. The diagonal elements are 4, 3, and 6. The trace is 4 + 3 + 6 = 13, and diag(A) = [[4, 0, 0], [0, 3, 0], [0, 0, 6]].

Applications include simplifying certain matrix operations, such as in diagonalization and when working with diagonal matrices that represent scaling along coordinate axes. Diagonalelementer are also important in numerical methods and data representations where only diagonal entries are nonzero or of interest, such as in sparse storage formats for diagonal matrices.