Diagonal Matrix

A diagonal matrix is a square matrix of the form

(1)

where is the Kronecker delta, are constants, and , 2, ..., , with no implied summation over indices. The general diagonal matrix is therefore of the form

(2)

often denoted .

The diagonal matrix with elements can be computed in the Wolfram Language using DiagonalMatrix[l], and a matrix may be tested to determine if it is diagonal using DiagonalMatrixQ[m].

The determinant of a diagonal matrix given by is . This means that , so for , 2, ..., the first few values are 1, 2, 6, 24, 120, 720, 5040, 40320, ... (OEIS A000142).

For matrices over a field, given a matrix equation of the form

(3)

multiply through to obtain

(4)

If the diagonal entries are pairwise distinct, this equation forces every off-diagonal entry of to vanish, so is diagonal. With repeated diagonal entries, a commuting matrix need not be diagonal.

Given a diagonal matrix and a positive integer , the matrix power is computed by taking each diagonal entry to that power,

			(5)
			(6)

The same formula holds for negative integer when all diagonal entries are nonzero. For a real or complex diagonal matrix, its matrix exponential is obtained by exponentiating the diagonal entries,

(7)