diagonaalmatrix

A diagonal matrix is a square matrix in which every non-diagonal entry is zero. Equivalently, a matrix A = [a_ij] is diagonal if a_ij = 0 for all i ≠ j. A diagonal matrix can be written as D = diag(d1, d2, ..., dn), where di are the entries on the main diagonal.

Properties and operations follow from its structure. The product of two diagonal matrices is diagonal, with diagonal entries equal to the products of the corresponding diagonal entries. Diagonal matrices commute with each other under multiplication. The inverse exists if and only if none of the diagonal entries is zero; in that case D^{-1} = diag(1/d1, 1/d2, ..., 1/dn). Powers are diagonal as well: D^k = diag(d1^k, d2^k, ..., dn^k). The determinant of a diagonal matrix is the product of its diagonal entries, and the trace is the sum of the diagonal entries. The transpose of a diagonal matrix is diagonal, and in particular a diagonal matrix is symmetric.

Eigenvalues and diagonalization are straightforward: the eigenvalues of a diagonal matrix are precisely its diagonal entries. Every diagonal matrix is already diagonalizable, and more generally a matrix is diagonalizable if it is similar to a diagonal matrix via A = P D P^{-1} for some invertible P.

Special cases include the identity matrix, which is diagonal with ones on the diagonal, and the zero matrix, which is diagonal with zeros on the diagonal. The collection of all n×n diagonal matrices forms a commutative subalgebra of the matrix algebra over a field. Diagonal matrices are widely used to represent scaling operations and to simplify linear algebra computations.