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A diagonal matrix is a square matrix of the form

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

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

multiply through to obtain

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,

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,

See also

Antisymmetric Matrix, Diagonal, Diagonal Quadratic Form, Diagonalizable Matrix, Hankel Matrix, Jordan Block, Matrix, Matrix Exponential, Normal Matrix, Symmetric Matrix, Triangular Matrix, Tridiagonal Matrix

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 181-184 and 217-229, 1985.Sloane, N. J. A. Sequence A000142/M1675 in "The On-Line Encyclopedia of Integer Sequences."

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Diagonal Matrix

Cite this as:

Weisstein, Eric W. "Diagonal Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DiagonalMatrix.html

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