It is thus a generalization of the scalar logarithm and in some sense an inverse function of the matrix exponential.
Because the exponential function is not bijective for complex numbers (e.g.
If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the power series which can be rewritten as Specifically, if
A rotation of angle α around the origin is represented by the 2×2-matrix For any integer n, the matrix is a logarithm of A.
This corresponds to the fact that the rotation angle is only determined up to multiples of 2π.
The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices.
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting.
A real matrix has a real logarithm if and only if it is invertible and each Jordan 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.
If A and B are both positive-definite matrices, then Suppose that A and B commute, meaning that AB = BA.
Setting B = A−1 in this equation yields Similarly, for non-commuting
can be obtained using the integral definition of the logarithm applied to both
The logarithm of such a rotation matrix R can be readily computed from the antisymmetric part of Rodrigues' rotation formula, explicitly in Axis angle.
It yields the logarithm of minimal Frobenius norm, but fails when R has eigenvalues equal to −1 where this is not unique.
A method for finding log A for a diagonalizable matrix A is the following: That the logarithm of A might be a complex matrix even if A 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 for rotation matrices).
The algorithm illustrated above does not work for non-diagonalizable matrices, such as For such matrices one needs to find its Jordan decomposition and, rather than computing the logarithm of diagonal entries as above, one would calculate the logarithm of the Jordan blocks.
The latter is accomplished by noticing that one can write a Jordan block as where K is a matrix with zeros on and under the main diagonal.
(The number λ is nonzero by the assumption that the matrix whose logarithm one attempts to take is invertible.)
Then, by the Mercator series one gets This series has a finite number of terms (Km is zero if m is equal to or greater than the dimension of K), and so its sum is well-defined.
Using this approach, one finds which can be verified by plugging the right-hand side into the matrix exponential:
Using the tools of holomorphic functional calculus, given a holomorphic function f defined on an open set in the complex plane and a bounded linear operator T, one can calculate f(T) as long as f is defined on the spectrum of T. The function f(z) = log z can be defined on any simply 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 ln T as long as the spectrum of T does not contain the origin and there is a path going from the origin to infinity not crossing the spectrum of T (e.g., if the spectrum of T is a circle with the origin inside of it, it is impossible to define ln T).
As long as the origin is not in the spectrum (the matrix is invertible), the path condition from the previous paragraph is satisfied, and ln T is well-defined.
Note that the exponential map is a local diffeomorphism between a neighborhood U of the zero matrix
[7] Thus the (matrix) logarithm is well-defined as a map, An important corollary of Jacobi's formula then is If a 2 × 2 real matrix has a negative determinant, it has no real logarithm.
Note first that any 2 × 2 real matrix can be considered one of the three types of the complex number z = x + y ε, where ε2 ∈ { −1, 0, +1 }.
This z is a point on a complex subplane of the ring of matrices.
The other three quadrants are images of this one under the Klein four-group generated by ε and −1.
It also follows, that, e.g., a square root of this matrix A is obtainable directly from exponentiating (logA)/2, For a richer example, start with a Pythagorean triple (p,q,r) and let a = log(p + r) − log q.