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.

This example computes and compares the matrix exponential of `A`

and
the exponential of `A`

.

A = [1 1 0 0 0 2 0 0 -1 ]; expm(A) ans = 2.7183 1.7183 1.0862 0 1.0000 1.2642 0 0 0.3679 exp(A) ans = 2.7183 2.7183 1.0000 1.0000 1.0000 7.3891 1.0000 1.0000 0.3679

Notice that the diagonal elements of the two results are equal. This would be true for any triangular matrix. But the off-diagonal elements, including those below the diagonal, are different.

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

[2] 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.

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

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