Wikipedia / Logarithm of a matrix
Wikipedia / Logarithm of a matrix
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 is defined by
.
is said to be a matrix logarithm.
Because the exponential function is not bijective for complex numbers
Power series expression
is sufficiently close to the identity matrix may be computed by means of the power series
,
which can be rewritten as
.
.
Example: Logarithm of rotations in the plane
matrix
, the matrix
.
where
...
qed.
.
are elements of the Lie group SO(2) are elements of the Lie algebra, which consists of all skew-symmetric matrices. The matrix
is a generator of this Lie algebra.
Existence
The question of whether a matrix has a logarithm has the easiest answer when considered in the complex setting. A complex matrix has a logarithm if and only if it is invertible. The logarithm is not unique, but if a matrix has no negative real eigenvalues. This logarithm is known as the principal logarithm.
The answer is more involved in the real 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. matrices is considered in a later section.
Properties
are both positive-definite matrices, then
. Then
is an eigenvalue is the corresponding eigenvalue. commute and are both positive-definite in this equation yields
, one can show that
can be obtained using the integral definition of the logarithm
.
Further example: Logarithm of rotations in 3D space
can be readily computed from the antisymmetric part of Rodrigues' rotation formula, explicitly in Axis angle. It yields the logarithm of minimal Frobenius norm where this is not unique.
,
manifold of rotation matrices.
Calculating the logarithm of a diagonalizable matrix
for a diagonalizable matrix is the following:
of eigenvectors).
Find the inverse.
Let
will be a diagonal matrix.
.
Then
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 non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.
Logarithm of a non-diagonalizable matrix
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 blockss.
The latter is accomplished by noticing that one can write a Jordan block as
is a matrix with zeros on and under the main diagonal 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), and so its sum is well-defined.
Example. Using this approach, one finds
A functional analysis perspective
A square matrix represents a linear operator on the Euclidean space is the dimension of the matrix. Since such a space is finite-dimensional, this operator is actually bounded.
Using the tools of holomorphic functional calculus, given a holomorphic function defined on an open set in the complex plane is defined on the spectrum.
can be defined on any simply connected).
is the set of eigenvalues of its matrix, and so is a finite set 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.
A Lie group theory perspective
In the theory of Lie groups, there is an exponential map from a Lie algebra
are square matrices and the exponential map is given by the matrix exponential. Note that the exponential map is a local diffeomorphism of the zero matrix. Thus the (matrix) logarithm is well-defined as a map,
An important corollary of Jacobi's formula then is
Constraints in the 2 × 2 case
real matrix has a negative determinant is a point on a complex subplane of the ring of matrices.
, that is a split-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 the Klein four-group
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.
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., a square root of this matrix A,
For a richer example, start with a Pythagorean triple Then
.
Now
.
Thus
has the logarithm matrix
,
See also
Notes
References
.
.
.