Inmathematics, alogarithm of a matrix is anothermatrix such that thematrix exponential of the latter matrix equals the original matrix. It is thus a generalization of the scalarlogarithm and in some sense aninverse function of thematrix 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 toLie theory since when a matrix has a logarithm then it is in an element of aLie group and the logarithm is the corresponding element of the vector space of theLie algebra.

Theexponential of a matrixA is defined by

${\displaystyle e^A\equiv \sum _{n=0}^{\mathrm{\infty }}\frac{A^n}{n!}}$.

Given a matrixB, another matrixA is said to be amatrix logarithm ofB ife^A =B.

Because the exponential function is notbijective forcomplex numbers (e.g.${\displaystyle e^{\pi i}=e^{3\pi i}=-1}$), 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${\displaystyle B}$ exists and is unique, then it is written as${\displaystyle \log B,}$ in which case${\displaystyle e^{\log B}=B.}$

IfB is sufficiently close to the identity matrix, then a logarithm ofB may be computed by means of thepower series

${\displaystyle \log (B)=\log (I+(B-I))=\sum _{k=1}^{\mathrm{\infty }}\frac{(-1{)}^{k+1}}{k}(B-I{)}^k=(B-I)-\frac{(B-I{)}^2}{2}+\frac{(B-I{)}^3}{3}-\cdots }$,

which can be rewritten as

${\displaystyle \log (B)=-\sum _{k=1}^{\mathrm{\infty }}\frac{(I-B{)}^k}{k}=-(I-B)-\frac{(I-B{)}^2}{2}-\frac{(I-B{)}^3}{3}-\cdots }$.

Specifically, if, then the preceding series converges and${\displaystyle e^{\log (B)}=B}$.^[1]

The rotations in the plane give a simple example. A rotation of angleα around the origin is represented by the 2×2-matrix

For any integern, the matrix

is a logarithm ofA.

Thus, the matrixA has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2π.

In the language of Lie theory, the rotation matricesA are elements of the Lie groupSO(2). The corresponding logarithmsB are elements of the Lie algebra so(2), which consists of allskew-symmetric matrices. The matrix

is a generator of theLie algebra so(2).

The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithmif and only if it isinvertible.^[2] The logarithm is not unique, but if a matrix has no negative realeigenvalues, then there is a unique logarithm that has eigenvalues all lying in the strip. This logarithm is known as theprincipal logarithm.^[3]

The answer is more involved in the real setting. A real matrix has a real logarithm if and only if it is invertible and eachJordan block belonging to a negative eigenvalue occurs an even number of times.^[4] If an invertible real matrix does not satisfy the condition with the Jordan blocks, then it has only non-real logarithms. This can already be seen in the scalar case: no branch of the logarithm can be real at -1. The existence of real matrix logarithms of real 2×2 matrices is considered in a later section.

IfA andB are bothpositive-definite matrices, then

${\displaystyle \mathrm{tr}\log (AB)=\mathrm{tr}\log (A)+\mathrm{tr}\log (B).}$

Suppose thatA andB commute, meaning thatAB =BA. Then

${\displaystyle \log (AB)=\log (A)+\log (B)}$

if and only if${\displaystyle \arg ({\mu }_j)+\arg ({\nu }_j)\in (-\pi ,\pi ]}$, where${\displaystyle {\mu }_j}$ is aneigenvalue of${\displaystyle A}$ and${\displaystyle {\nu }_j}$ is the correspondingeigenvalue of${\displaystyle B}$.^[5] In particular,${\displaystyle \log (AB)=\log (A)+\log (B)}$ whenA andB commute and are bothpositive-definite. SettingB =A⁻¹ in this equation yields

${\displaystyle \log (A^{-1})=-\log (A).}$

Similarly, for non-commuting${\displaystyle A}$ and${\displaystyle B}$, one can show that^[6]

${\displaystyle \log (A+tB)=\log (A)+t{\int }_0^{\mathrm{\infty }}dz\frac{I}{A+zI}B\frac{I}{A+zI}+O(t^2).}$

More generally, a series expansion of${\displaystyle \log (A+tB)}$ in powers of${\displaystyle t}$ can be obtained using the integral definition of the logarithm

${\displaystyle \log (X+\lambda I)-\log (X)={\int }_0^{\lambda }dz\frac{I}{X+zI},}$

applied to both${\displaystyle X=A}$ and${\displaystyle X=A+tB}$ in the limit${\displaystyle \lambda \to \mathrm{\infty }}$.

A rotationR ∈ SO(3) in${\displaystyle \mathbb{R}}$³ is given by a 3×3orthogonal matrix.

The logarithm of such a rotation matrixR can be readily computed from the antisymmetric part ofRodrigues' rotation formula, explicitly inAxis angle. It yields the logarithm of minimalFrobenius norm, but fails whenR has eigenvalues equal to −1 where this is not unique.

Further note that, given rotation matricesA andB,

${\displaystyle d_g(A,B):=\Vert \log (A^{\text{T}}B){\Vert }_F}$

is the geodesic distance on the 3D manifold of rotation matrices.

A method for finding logA for adiagonalizable matrixA is the following:

Find the matrixV ofeigenvectors ofA (each column ofV is an eigenvector ofA).

Find theinverseV⁻¹ ofV.

Let

${\displaystyle A^{\prime }=V^{-1}AV.}$

ThenA′ will be adiagonal matrix whose diagonal elements are eigenvalues ofA.

Replace each diagonal element ofA′ by its (natural) logarithm in order to obtain${\displaystyle \log A^{\prime }}$.

Then

${\displaystyle \log A=V(\log A^{\prime })V^{-1}.}$

That the logarithm ofA might be a complex matrix even ifA is real then follows from the fact that a matrix with real and positive entries might nevertheless have negative or even complex eigenvalues (this is true for example forrotation matrices). The non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.

The algorithm illustrated above does not work for non-diagonalizable matrices, such as

For such matrices one needs to find itsJordan decomposition and, rather than computing the logarithm of diagonal entries as above, one would calculate the logarithm of theJordan blocks.

The latter is accomplished by noticing that one can write a Jordan block as

whereK is a matrix with zeros on and under themain diagonal. (The number λ is nonzero by the assumption that the matrix whose logarithm one attempts to take is invertible.)

Then, by theMercator series

${\displaystyle \log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots }$

one gets

${\displaystyle \log B=\log (\lambda (I+K))=\log (\lambda I)+\log (I+K)=(\log \lambda )I+K-\frac{K^2}{2}+\frac{K^3}{3}-\frac{K^4}{4}+\cdots }$

Thisseries has a finite number of terms (K^m is zero ifm is equal to or greater than the dimension ofK), and so its sum is well-defined.

Example. Using this approach, one finds

which can be verified by plugging the right-hand side into the matrix exponential:

Another, more standard, method by theHermite interpolation with the use ofconfluent Vandermonde matrix is presented by Higham.^[7]

A square matrix represents alinear operator on theEuclidean spaceRⁿ wheren is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actuallybounded.

Using the tools ofholomorphic functional calculus, given aholomorphic functionf defined on anopen set in thecomplex plane and a bounded linear operatorT, one can calculatef(T) as long asf is defined on thespectrum ofT.

The functionf(z) = logz can be defined on anysimply connected open set in the complex plane not containing the origin, and it is holomorphic on such a domain. This implies that one can define lnT as long as the spectrum ofT does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum ofT (e.g., if the spectrum ofT is a circle with the origin inside of it, it is impossible to define lnT).

The spectrum of a linear operator onRⁿ is the set of eigenvalues of its matrix, and so is a finite set. As long as the origin is not in the spectrum (the matrix is invertible), the path condition from the previous paragraph is satisfied, and lnT is well-defined. The non-uniqueness of the matrix logarithm follows from the fact that one can choose more than one branch of the logarithm which is defined on the set of eigenvalues of a matrix.

In the theory ofLie groups, there is anexponential map from aLie algebra${\displaystyle \mathfrak{g}}$ to the corresponding Lie groupG

${\displaystyle \exp :\mathfrak{g}\to G.}$

For matrix Lie groups, the elements of${\displaystyle \mathfrak{g}}$ andG are square matrices and the exponential map is given by thematrix exponential. The inverse map${\displaystyle \log ={\exp }^{-1}}$ is multivalued and coincides with the matrix logarithm discussed here. The logarithm maps from the Lie groupG into the Lie algebra${\displaystyle \mathfrak{g}}$. Note that the exponential map is alocal diffeomorphism between a neighborhoodU of the zero matrix${\displaystyle \underset{\_}{0}\in \mathfrak{g}}$ and a neighborhoodV of the identity matrix${\displaystyle \underset{\_}{1}\in G}$.^[8]Thus the (matrix) logarithm is well-defined as a map,

${\displaystyle \log :G\supset V\to U\subset \mathfrak{g}.}$

An important corollary ofJacobi's formula then is

${\displaystyle \log (det(A))=\mathrm{t}\mathrm{r}(\log A).}$

If a 2 × 2 real matrix has a negativedeterminant, it has no real logarithm. Note first that any 2 × 2 real matrix can be considered one of the three types of the complex numberz =x +yε, where ε² ∈ { −1, 0, +1 }. Thisz is a point on a complex subplane of thering of matrices.^[9]

The case where the determinant is negative only arises in a plane with ε² =+1, that is asplit-complex number plane. Only one quarter of this plane is the image of the exponential map, so the logarithm is only defined on that quarter (quadrant). The other three quadrants are images of this one under theKlein four-group generated by ε and −1.

For example, leta = log 2 ; then cosha = 5/4 and sinha = 3/4.For matrices, this means that

.

So this last matrix has logarithm

.

These matrices, however, do not have a logarithm:

.

They represent the three other conjugates by the four-group of the matrix above that does have a logarithm.

A non-singular 2 × 2 matrix does not necessarily have a logarithm, but it is conjugate by the four-group to a matrix that does have a logarithm.

It also follows, that, e.g., asquare root of this matrixA is obtainable directly from exponentiating (logA)/2,

For a richer example, start with aPythagorean triple (p,q,r)and leta = log(p +r) − logq. Then

${\displaystyle e^a=\frac{p+r}{q}=\cosh a+\sinh a}$.

Now

.

Thus

has the logarithm matrix

,

wherea = log(p +r) − logq.

• Matrix function
• Square root of a matrix
• Matrix exponential
• Baker–Campbell–Hausdorff formula
• Derivative of the exponential map

• Gantmacher, Felix R. (1959),The Theory of Matrices, vol. 1, New York: Chelsea, pp. 239–241.
• Hall, Brian C. (2015),Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, vol. 222 (2nd ed.), Springer,ISBN 978-3319134666
• Culver, Walter J. (1966), "On the existence and uniqueness of the real logarithm of a matrix",Proceedings of the American Mathematical Society,17 (5):1146–1151,doi:10.1090/S0002-9939-1966-0202740-6,ISSN 0002-9939.
• Higham, Nicholas (2008),Functions of Matrices. Theory and Computation,SIAM,ISBN 978-0-89871-646-7.
• Engø, Kenth (June 2001),"On the BCH-formula inso(3)",BIT Numerical Mathematics,41 (3):629–632,doi:10.1023/A:1021979515229,ISSN 0006-3835,S2CID 126053191

• Matrix theory
• Inverse functions
• Logarithms

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