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.
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How does the difference between point 0-0-0-0-0-0-0-3-9-8 and point 0-0-0-0-0-0-0-0-3-9-8 cause one to have red eyes after swimming? To answer this, we first need a way of dealing with rather small numbers, or in some cases extremely large numbers. This leads us to the concept of logarithms. Well, what are logarithms? Let's take the base number - b - and raise it to a power, p, like 2 to the 3rd power and have it equal a number n. We get an exponential equation b raised to the p power equals n. In our example, that'd be 2 raised to the 3rd power equals 8. The exponent p is said to be the logarithm of the number n. Most of the time this would be written "log base b of a number equals p, the power." This is starting to sound a bit confusing with all the variables, so let's show this with an example. What is the value of log base 10 of 10 thousand? The same question could be asked using exponents. 10 raised to what power is 10 thousand? Well, 10 to the 4th is 10 thousand. So, log base 10 of 10 thousand must equal 4. This example can also be completed very simply on a scientific calculator. Log base 10 is used so frequently in the sciences that it has the honor of having its own button on most calculators. If the calculator will figure out logs for me, why study them? Just a quick reminder, the log button only computes logarithms of base 10. What if you want to go into computer science and need to understand base 2? So what is log base 2 of 64? In other words, 2 raised to what power is 64? Well, use your fingers. 2, 4, 8, 16, 32, 64. So log base 2 of 64 must equal 6. So what does this have to do with my eyes turning red in some swimming pools and not others? Well, it leads us into an interesting use of logarithms in chemistry: finding the pH of water samples. pH tells us how acidic or basic a sample is, and can be calculated with the formula pH equals negative log base 10 of the hydrogen ion concentration, or H plus. We can find the pH of water samples with hydrogen ion concentration of point 0-0-0-0-0-0-0-3-9-8 and point 0-0-0-0-0-0-0-0-3-9-8 quickly on a calculator. Punch: negative log of each of those numbers, and you'll see the pHs are 7.4 and 8.4. Since the tears in our eyes have a pH of about 7.4, the H plus concentration of .70398 will feel nice on your eyes. But the pH of 8.4 will make you feel itchy and red. It's easy to remember logarithms - log base b of some number n equals p - by repeating "the base raised to what power equals the number?" The base raised to what power equals the number? The base raised to what power equals the number? So now we know logarithms are very powerful when dealing with extremely small or large numbers. Logarithms can even be used instead of eyedrops after swimming.
Definition
The exponential of a matrix A is defined by
- .
Given a matrix B, another matrix A is said to be a matrix logarithm of B if eA = B.
Because the exponential function is not bijective for complex numbers (e.g. ), 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 exists and is unique, then it is written as in which case
Power series expression
If B is sufficiently close to the identity matrix, then a logarithm of B may be computed by means of the following power series:
- .
Specifically, if , then the preceding series converges and .[1]
Example: Logarithm of rotations in the plane
The rotations in the plane give a simple example. 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.
Proof
|
---|
⇔ where
|
Thus, the matrix A 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 matrices A are elements of the Lie group SO(2). The corresponding logarithms B are elements of the Lie algebra so(2), which consists of all skew-symmetric matrices. The matrix
is a generator of the Lie algebra so(2).
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.[2] The logarithm is not unique, but if a matrix has no negative real eigenvalues, then there is a unique logarithm that has eigenvalues all lying in the strip . This logarithm is known as the principal 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 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. The existence of real matrix logarithms of real 2×2 matrices is considered in a later section.
Properties
If A and B are both positive-definite matrices, then
Suppose that A and B commute, meaning that AB = BA. Then
if and only if , where is an eigenvalue of and is the corresponding eigenvalue of .[5] In particular, when A and B commute and are both positive-definite. Setting B = A−1 in this equation yields
Similarly, for non-commuting and , one can show that[6]
More generally, a series expansion of in powers of can be obtained using the integral definition of the logarithm
applied to both and in the limit .
Further example: Logarithm of rotations in 3D space
A rotation R ∈ SO(3) in 3 is given by a 3×3 orthogonal matrix.
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.
Further note that, given rotation matrices A and B,
is the geodesic distance on the 3D manifold of rotation matrices.
Calculating the logarithm of a diagonalizable matrix
A method for finding log A for a diagonalizable matrix A is the following:
- Find the matrix V of eigenvectors of A (each column of V is an eigenvector of A).
- Find the inverse V−1 of V.
- Let
- Then A′ will be a diagonal matrix whose diagonal elements are eigenvalues of A.
- Replace each diagonal element of A′ by its (natural) logarithm in order to obtain .
- Then
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 non-uniqueness of the logarithm of a matrix follows from the non-uniqueness of the logarithm of a complex number.
The 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 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.
Example. Using this approach, one finds
which can be verified by plugging the right-hand side into the matrix exponential:
A functional analysis perspective
A square matrix represents a linear operator on the Euclidean space Rn where n 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 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).
The spectrum of a linear operator on Rn 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 ln T 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 to the corresponding Lie group G
For matrix Lie groups, the elements of and G are square matrices and the exponential map is given by the matrix exponential. The inverse map is multivalued and coincides with the matrix logarithm discussed here. The logarithm maps from the Lie group G into the Lie algebra . Note that the exponential map is a local diffeomorphism between a neighborhood U of the zero matrix and a neighborhood V of the identity matrix .[7] 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
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.[8]
The case where the determinant is negative only arises in a plane with ε2 =+1, 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 generated by ε and −1.
For example, let a = log 2 ; then cosh a = 5/4 and sinh a = 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 x 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., 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. Then
- .
Now
- .
Thus
has the logarithm matrix
- ,
where a = log(p + r) − log q.
See also
- Matrix function
- Square root of a matrix
- Matrix exponential
- Baker–Campbell–Hausdorff formula
- Derivative of the exponential map
Notes
- ^ Hall 2015 Theorem 2.8
- ^ Higham (2008), Theorem 1.27
- ^ Higham (2008), Theorem 1.31
- ^ Culver (1966)
- ^ APRAHAMIAN, MARY; HIGHAM, NICHOLAS J. (2014). "The Matrix Unwinding Function, with an Application to Computing the Matrix Exponential". SIAM Journal on Matrix Analysis and Applications. 35 (1): 97. doi:10.1137/130920137. Retrieved 13 December 2022.
- ^ Unpublished memo by S Adler (IAS)
- ^ Hall 2015 Theorem 3.42
- ^
References
- 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 in so(3)", BIT Numerical Mathematics, 41 (3): 629–632, doi:10.1023/A:1021979515229, ISSN 0006-3835, S2CID 126053191