diagonalmatrix

A diagonal matrix is a square matrix with zeros in all off-diagonal positions; only the entries on the main diagonal may be nonzero. It is often written as D = diag(d1, ..., dn), where the numbers d1 through dn are the diagonal entries. In index form, D_{ij} = 0 if i ≠ j and D_{ii} = d_i.

Key properties follow directly from the definition. The determinant of a diagonal matrix is the product of its diagonal entries: det(D) = d1 d2 ... dn. The inverse exists precisely when all diagonal entries are nonzero, in which case D^{-1} = diag(1/d1, ..., 1/dn). If D and E are diagonal matrices with diagonal entries diag(d1, ..., dn) and diag(e1, ..., en), then their product is diagonal: DE = diag(d1 e1, ..., dn en). Diagonal matrices commute with each other, and they are generally not commuting with arbitrary non-diagonal matrices.

Eigenvalues and eigenvectors are straightforward: the eigenvalues of D are the diagonal entries d1, ..., dn, with corresponding eigenvectors being the standard basis vectors e1, ..., en. Consequently, a diagonal matrix represents a linear transformation that scales each coordinate axis by the respective diagonal entry.

Special cases and related concepts include scalar matrices cI, which are diagonal with equal diagonal entries, and the identity matrix I, which is the diagonal matrix with all ones. Diagonal matrices are inherently simple to work with and serve as a canonical form in many contexts. A matrix is diagonalizable if it is similar to a diagonal matrix; in that case, there exists an invertible P such that P^{-1}AP = D is diagonal. Diagonal matrices appear across linear algebra in solving systems, performing powers, and reducing more complex matrices via similarity transformations.