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MatrixLog: Matrix natural logarithm —Wolfram Documentation

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MatrixLog[m]

gives the matrix logarithm of a matrix m.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
Applications  
Properties & Relations  
Possible Issues  
Neat Examples  
See Also
Related Guides
History
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MatrixLog

MatrixLog[m]

gives the matrix logarithm of a matrix m.

Details and Options

  • MatrixLog is effectively the functional inverse of MatrixExp, so that MatrixExp[MatrixLog[m]] is m for a nonsingular matrix.
  • MatrixLog works only on square nonsingular matrices.
  • MatrixLog works on SparseArray objects and structured arrays.
  • The following option can be given:
  • Method Automaticmethod to use
  • Possible values for the option Method include:
  • Automaticautomatic selection
    "Jordan"Jordan decomposition
    "Schur"Schur decomposition for inexact numerical matrices
    "SpectralBasis"spectral method for exact matrices »
  • The default setting of Method->Automatic uses the Schur decomposition for inexact numerical matrices and, based on matrix contents, selects between the Jordan method and the spectral basis method for exact and symbolic matrices.
  • The "Schur" method can also be specified by Method->{"Schur",Tolerance->tol}, where eigenvalues with a relative magnitude less than tol are effectively considered as 0 so that the matrix is considered as singular. Use Tolerance->0 to allow any nonzero eigenvalue, no matter how small.

Examples

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Basic Examples  (1)

Logarithm of a 2×2 matrix:

Wolfram Language code: MatrixLog[{{6.557, 1.734}, {4.523, 7.425}}]
Wolfram Language code: MatrixLog[{{2, 1}, {1, 2}} t]//Simplify

Scope  (4)

Use exact arithmetic to compute the matrix logarithm:

Wolfram Language code: m = {{1, 1 / 2, 1 / 3, 1 / 4}, {0, 1, 1 / 3, 1 / 4}, {0, 0, 1, 1 / 4}, {0, 0, 0, 1}};
Wolfram Language code: MatrixLog[m]//MatrixForm

Use machine arithmetic:

Wolfram Language code: MatrixLog[N[m]]//MatrixForm

Use 24-digit-precision arithmetic:

Wolfram Language code: MatrixLog[N[m, 24]]

Compute a matrix logarithm of a complex matrix:

Wolfram Language code: m = RandomComplex[1 + I, {3, 3}];
Wolfram Language code: MatrixLog[m]//MatrixForm

A matrix logarithm of a symbolic matrix:

Wolfram Language code: m = {{E, 0, 0}, {0, 4 E, -4 E}, {0, 4 E, 4 E}};
Wolfram Language code: MatrixLog[m t]//FullSimplify[#, t > 0]&//MatrixForm

A matrix logarithm of a sparse 10×10 matrix applied to a vector:

Wolfram Language code: MatrixLog[SparseArray[{Band[{1, 1}] -> 2, Band[{1, 2}] -> -1}, {10, 10}]].Range[10]

Options  (4)

Method  (4)

The method "Jordan" can work with exact and inexact matrices:

Wolfram Language code: MatrixLog[{{2, 1}, {3, 2}}, Method -> "Jordan"]//Simplify
Wolfram Language code: MatrixLog[{{2., 1.}, {3., 2.}}, Method -> "Jordan"]//Chop

The method "Schur" works only with inexact matrices:

Wolfram Language code: MatrixLog[{{2, 1}, {3, 2}}, Method -> "Schur"]
Wolfram Language code: MatrixLog[{{2., 1.}, {3., 2.}}, Method -> "Schur"]//Chop

Eigenvalues with relatively very small magnitude are treated as zero:

Wolfram Language code: m = {{10.^10, 10.}, {0., 2 10.^-4}}; MatrixLog[m]

Use Tolerance->0 to include nonzero eigenvalues of any magnitude:

Wolfram Language code: MatrixLog[m, Method -> {"Schur", Tolerance -> 0}]

When it is applicable, "SpectralBasis" can be orders of magnitude faster than "Jordan":

Wolfram Language code: mat = {{0, 0, 0, -2, -2, 0, 2, -1}, {0, -2, 4, 2, -2, 2, 5, 2}, {4, 0, 0, 2, 4, -3, -2, 6}, {-2, -2, 2, -2, -1, -4, 0, 2}, {0, -2, 2, -1, 0, 2, -2, -2}, {0, -4, 1, 2, -2, 2, -4, 0}, {0, 1, 0, 0, 2, 0, -4, 0}, {-1, 0, -2, 0, 0, 0, -4, -2}}; AbsoluteTiming[mexp1 = MatrixLog[mat, Method -> "SpectralBasis"];]
Wolfram Language code: AbsoluteTiming[mexp2 = MatrixLog[mat, Method -> "Jordan"];]

It can also produce much simpler results:

Wolfram Language code: AbsoluteTiming[LeafCount[mexp1]]
Wolfram Language code: AbsoluteTiming[LeafCount[mexp2]]

The two results are equal, however:

Wolfram Language code: AbsoluteTiming[Chop[N[mexp1 - mexp2]]]

The method is chosen automatically for matrices of exact numbers, where it excels:

Wolfram Language code: MatrixQ[mat, NumberQ] && Precision[mat] == Infinity && mexp1 === MatrixLog[mat]

Applications  (1)

Given the vectors , and a number , compute the matrix for which :

Wolfram Language code: n = 10; h = 1 / 10; z = NestList[RotateLeft, Range[n], n]//N; TableForm[{z}, TableHeadings -> {None, Range[0, n]}]

Form the matrices s=TemplateBox[{{{, {{z, _, 0}, ,, ..., ,, {z, _, {(, {n, -, 1}, )}}}, }}}, Transpose] and w=TemplateBox[{{{, {{z, _, 1}, ,, ..., ,, {z, _, n}}, }}}, Transpose] to compute the matrix such that :

Wolfram Language code: s = Take[z, {1, -2}]; w = Take[z, {2, -1}]; A = MatrixLog[w.Inverse[s]] / h;

Check that applying the matrix iteratively starting with generates the rest of the vectors :

Wolfram Language code: res = NestList[MatrixExp[A h].#&, z[[1]], n]//Chop; TableForm[{res}, TableHeadings -> {None, Range[0, n]}]

Properties & Relations  (2)

If m is a nonsingular numeric matrix, then MatrixExp[MatrixLog[m]] is effectively equal to m:

Wolfram Language code: m = {{0, 0, 1}, {0, 2, 0}, {-1, 0, 0}}; mm = MatrixExp[MatrixLog[m]]; m == mm

But it is not always true for symbolic matrices:

Wolfram Language code: m = {{0, t}, {-t, -2 t}}; MatrixExp[MatrixLog[m]]

The identity does not hold in general:

Wolfram Language code: m = {{6, 0, 10}, {0, 20, 0}, {-2, 0, 0}}; mm = MatrixLog[MatrixExp[m]]//Simplify; m - mm//FullSimplify

If a matrix is positive definite, then the identity holds:

Wolfram Language code: m = {{2, 0, 5}, {0, 1 / 2, 0}, {-1, 0, 7}}; mm = MatrixLog[MatrixExp[m]]//Simplify; m - mm//FullSimplify
Wolfram Language code: Eigenvalues[m]//N

Possible Issues  (1)

MatrixLog does not work with singular matrices:

Wolfram Language code: MatrixLog[{{0, 0, 1}, {0, 0, 0}, {-1, 0, 0}}]
Wolfram Language code: MatrixLog[{{0., 0., 1.}, {0., 0., 0.}, {-1., 0., 0.}}]

Neat Examples  (1)

Construct a cross from the reversed identity matrix:

Wolfram Language code: m = Reverse@IdentityMatrix[11]; FullSimplify[MatrixLog[m]]//MatrixForm//IconizedObject[«make fractions big»]
Wolfram Research (2012), MatrixLog, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixLog.html (updated 2026).

Text

Wolfram Research (2012), MatrixLog, Wolfram Language function, https://reference.wolfram.com/language/ref/MatrixLog.html (updated 2026).

CMS

Wolfram Language. 2012. "MatrixLog." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2026. https://reference.wolfram.com/language/ref/MatrixLog.html.

APA

Wolfram Language. (2012). MatrixLog. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MatrixLog.html

BibTeX

@misc{reference.wolfram_2026_matrixlog, author="Wolfram Research", title="{MatrixLog}", year="2026", howpublished="\url{https://reference.wolfram.com/language/ref/MatrixLog.html}", note=[Accessed: 14-November-2026]}

BibLaTeX

@online{reference.wolfram_2026_matrixlog, organization={Wolfram Research}, title={MatrixLog}, year={2026}, url={https://reference.wolfram.com/language/ref/MatrixLog.html}, note=[Accessed: 14-November-2026]}