This is machine translation

Translated by Microsoft
Mouseover text to see original. Click the button below to return to the English version of the page.

Note: This page has been translated by MathWorks. Click here to see
To view all translated materials including this page, select Country from the country navigator on the bottom of this page.


Matrix exponential


Y = expm(X)



Y = expm(X) computes the matrix exponential of X. Although it is not computed this way, if X has a full set of eigenvectors V with corresponding eigenvalues D, then [V,D] = eig(X) and

expm(X) = V*diag(exp(diag(D)))/V

Use exp for the element-by-element exponential.


collapse all

Compute and compare the exponential of A with the matrix exponential of A.

A = [1 1 0; 0 0 2; 0 0 -1];
ans = 3×3

    2.7183    2.7183    1.0000
    1.0000    1.0000    7.3891
    1.0000    1.0000    0.3679

ans = 3×3

    2.7183    1.7183    1.0862
         0    1.0000    1.2642
         0         0    0.3679

Notice that the diagonal elements of the two results are equal, which is true for any triangular matrix. The off-diagonal elements, including those below the diagonal, are different.

Input Arguments

collapse all

Input matrix, specified as a square matrix.

Data Types: single | double
Complex Number Support: Yes


The algorithm expm uses is described in [1] and [2].


The files, expmdemo1.m, expmdemo2.m, and expmdemo3.m illustrate the use of Padé approximation, Taylor series approximation, and eigenvalues and eigenvectors, respectively, to compute the matrix exponential. References [3] and [4] describe and compare many algorithms for computing a matrix exponential.


[1] Higham, N. J., “The Scaling and Squaring Method for the Matrix Exponential Revisited,” SIAM J. Matrix Anal. Appl., 26(4) (2005), pp. 1179–1193.

[2] Al-Mohy, A. H. and N. J. Higham, “A new scaling and squaring algorithm for the matrix exponential,” SIAM J. Matrix Anal. Appl., 31(3) (2009), pp. 970–989.

[3] Golub, G. H. and C. F. Van Loan, Matrix Computation, p. 384, Johns Hopkins University Press, 1983.

[4] Moler, C. B. and C. F. Van Loan, “Nineteen Dubious Ways to Compute the Exponential of a Matrix,” SIAM Review 20, 1978, pp. 801–836. Reprinted and updated as “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later,” SIAM Review 45, 2003, pp. 3–49.

Extended Capabilities

See Also

| | | | |

Introduced before R2006a