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.
A Method option can be given, with possible explicit settings:
"Jordan" Jordan decomposition

"Schur" Schur decomposition for inexact numerical matrices
The default setting of Method->Automatic uses the Schur decomposition for inexact numerical matrices and Jordan decomposition 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.
MatrixLog works on SparseArray objects.

Examples

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Basic Examples (1)

Logarithm of a 2×2 matrix:

Scope (4)

Use exact arithmetic to compute the matrix logarithm:

Use machine arithmetic:

Use 24-digit-precision arithmetic:

Compute a matrix logarithm of a complex matrix:

A matrix logarithm of a symbolic matrix:

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

Options (1)

Method (1)

Eigenvalues with relatively very small magnitude are treated as zero:

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

Applications (1)

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

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 :