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Diagonal Matrix


A diagonal matrix is a square matrix A of the form

 a_(ij)=c_idelta_(ij),
(1)

where delta_(ij) is the Kronecker delta, c_i are constants, and i,j=1, 2, ..., n, with no implied summation over indices. The general diagonal matrix is therefore of the form

 [c_1 0 ... 0; 0 c_2 ... 0; | | ... |; 0 0 ... c_n],
(2)

often denoted diag(c_1,c_2,...,c_n).

The diagonal matrix with elements l={c_1,...,c_n} can be computed in the Wolfram Language using DiagonalMatrix[l], and a matrix m may be tested to determine if it is diagonal using DiagonalMatrixQ[m].

The determinant of a diagonal matrix given by diag(c_1,c_2,...,c_n) is c_1c_2...c_n. This means that diag(1,2,...,n)=n!, so for n=1, 2, ..., the first few values are 1, 2, 6, 24, 120, 720, 5040, 40320, ... (OEIS A000142).

Given a matrix equation of the form

 [a_(11) ... a_(1n); | ... |; a_(n1) ... a_(nn)][lambda_1 ... 0; | ... |; 0 ... lambda_n]=[lambda_1 ... 0; | ... |; 0 ... lambda_n][a_(11) ... a_(1n); | ... |; a_(n1) ... a_(nn)],
(3)

multiply through to obtain

 [a_(11)lambda_1 ... a_(1n)lambda_n; | ... |; a_(n1)lambda_1 ... a_(nn)lambda_n]=[a_(11)lambda_1 ... a_(1n)lambda_1; | ... |; a_(n1)lambda_n ... a_(nn)lambda_n].
(4)

Since in general, lambda_i!=lambda_j for i!=j, this can be true only if off-diagonal components vanish. Therefore, A must be diagonal.

Given a diagonal matrix T, the matrix power can be computed simply by taking each element to the power in question,

T^n=[t_1 0 ... 0; 0 t_2 ... 0; | | ... |; 0 0 ... t_k]^n
(5)
=[t_1^n 0 ... 0; 0 t_2^n ... 0; | | ... |; 0 0 ... t_k^n].
(6)

Similarly, a matrix exponential can be performed simply by exponentiating each of the diagonal elements,

 exp(T)=[e^(t_1) 0 ... 0; 0 e^(t_2) ... 0; | | ... |; 0 0 ... e^(t_k)].
(7)

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."

Referenced on Wolfram|Alpha

Diagonal Matrix

Cite this as:

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

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