In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential. Not all matrices have a logarithm and those matrices that do have a logarithm may have more than one logarithm. The study of logarithms of matrices leads to Lie theory since when a matrix has a logarithm then it is in an element of a Lie group and the logarithm is the corresponding element of the vector space of the Lie algebra.
Definition The exponential of a matrix A {\displaystyle A} is defined by
e A ≡ ∑ n = 0 ∞ A n n ! {\displaystyle e^{A}\equiv \sum _{n=0}^{\infty }{\frac {A^{n}}{n!}}} . Given a matrix B {\displaystyle B} , another matrix A {\displaystyle A} is said to be a matrix logarithm of B {\displaystyle B} if e A = B {\displaystyle e^{A}=B} . Because the exponential function is not bijective for complex numbers (e.g. e π i = e 3 π i = − 1 {\displaystyle e^{\pi i}=e^{3\pi i}=-1} ), numbers can have multiple complex logarithms, and as a consequence of this, some matrices may have more than one logarithm, as explained below. If the matrix logarithm of B {\displaystyle B} exists, then it is written as log B , {\displaystyle \log B,} in which case e log B = B . {\displaystyle e^{\log B}=B.}
Power series expression If B {\displaystyle B} is sufficiently close to the identity matrix, then a logarithm of B {\displaystyle B} may be computed by means of the power series
log ( B ) = log ( I + ( B − I ) ) = ∑ k = 1 ∞ ( − 1 ) k + 1 k ( B − I ) k = ( B − I ) − ( B − I ) 2 2 + ( B − I ) 3 3 − ⋯ {\displaystyle \log(B)=\log(I+(B-I))=\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}(B-I)^{k}=(B-I)-{\frac {(B-I)^{2}}{2}}+{\frac {(B-I)^{3}}{3}}-\cdots } , which can be rewritten as
log ( B ) = − ∑ k = 1 ∞ ( I − B ) k k = − ( I − B ) − ( I − B ) 2 2 − ( I − B ) 3 3 − ⋯ {\displaystyle \log(B)=-\sum _{k=1}^{\infty }{\frac {(I-B)^{k}}{k}}=-(I-B)-{\frac {(I-B)^{2}}{2}}-{\frac {(I-B)^{3}}{3}}-\cdots } . Specifically, if ‖ I − B ‖ < 1 {\displaystyle \left\|I-B\right\|<1} , then the preceding series converges and e log ( B ) = B {\displaystyle e^{\log(B)}=B} .
Example: Logarithm of rotations in the plane The rotations in the plane give a simple example. A rotation of angle α {\displaystyle \alpha } around the origin is represented by the 2 × 2 {\displaystyle 2\times 2} matrix
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