diagonalmatrise

In linear algebra, a diagonalmatrise, or diagonal matrix, is a square matrix in which all off-diagonal elements are zero. The diagonal entries may be any numbers, including zero. A common notation is D = diag(d1, d2, ..., dn), representing an n × n matrix with di on the i-th diagonal position and zeros elsewhere.

Diagonalmatrises have several simple and useful properties. Multiplying a vector x by D scales each component x_i by the corresponding diagonal entry d_i, so D x = (d1 x1, d2 x2, ..., dn x_n). Similarly, left- or right-multiplication by D scales rows or columns of another matrix. Powers of a diagonalmatrise are easy to compute: D^k = diag(d1^k, d2^k, ..., dn^k).

Key numerical characteristics include the determinant and trace. The determinant of a diagonalmatrise is the product of its diagonal entries, det(D) = ∏ d_i, and the trace is the sum of the diagonal entries, tr(D) = ∑ d_i. If every diagonal entry is nonzero, D is invertible, with inverse D^{-1} = diag(1/d1, 1/d2, ..., 1/dn). The eigenvalues of a diagonalmatrise are precisely its diagonal entries, and the corresponding eigenvectors are the standard basis vectors.

Diagonalmatrises are inherently diagonalizable, since they are already in diagonal form. More generally, a matrix A is diagonalizable if there exists an invertible P such that P^{-1} A P = D, where D is diagonal. Not all matrices possess this property.

Applications include simplifying linear transformations, computing powers and exponentials, and serving as a basis for understanding more complex matrices through diagonalization. Example: diag(2, -5, 7) is a 3×3 diagonalmatrise representing independent stretch factors along coordinate axes.