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Matrix logarithm



L = logm(A) is the principal matrix logarithm of A, the inverse of expm(A). L is the unique logarithm for which every eigenvalue has imaginary part lying strictly between -pi and pi. If A is singular or has any eigenvalues on the negative real axis, the principal logarithm is undefined. In this case, logm computes a non-principal logarithm and returns a warning message.

[L, exitflag] = logm(A) returns a scalar exitflag that describes the exit condition of logm:

The input A can have class double or single.


If A is real symmetric or complex Hermitian, then so is logm(A).

Some matrices, like A = [0 1; 0 0], do not have any logarithms, real or complex, so logm cannot be expected to produce one.


For most matrices:

These identities may fail for some A. For example, if the computed eigenvalues of A include an exact zero, then logm(A) generates infinity. Or, if the elements of A are too large, expm(A) may overflow.


Suppose A is the 3-by-3 matrix

and Y = expm(A) is

Then A = logm(Y) produces the original matrix A.

But log(A) involves taking the logarithm of zero, and so produces


The algorithm logm uses is described in [1].

See Also

expm, funm, sqrtm


[1]  Davies, P. I. and N. J. Higham, "A Schur-Parlett algorithm for computing matrix functions," SIAM J. Matrix Anal. Appl., Vol. 25, Number 2, pp. 464-485, 2003.

[2]  Cheng, S. H., N. J. Higham, C. S. Kenney, and A. J. Laub, "Approximating the logarithm of a matrix to specified accuracy," SIAM J. Matrix Anal. Appl., Vol. 22, Number 4, pp. 1112-1125, 2001.

[3]  Higham, N. J., "Evaluating Pade approximants of the matrix logarithm," SIAM J. Matrix Anal. Appl., Vol. 22, Number 4, pp. 1126-1135, 2001.

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

[5]  Moler, C. B. and C. F. Van Loan, "Nineteen Dubious Ways to Compute the Exponential of a Matrix," SIAM Review 20, 1979,pp. 801-836.

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