A th root of an matrix is a matrix such that , and it can be written . For a rational number (where and are integers), defining is more difficult: is it or ? These two possibilities can be different even for . More generally, how can we define for an arbitrary real number ?
Recall, first, that for a nonzero complex scalar we define , where is the principal logarithm: the one taking values in the strip . We can generalize this definition to matrices. For a nonsingular matrix we define
Here the logarithm is the principal matrix logarithm, the matrix function built on the principal scalar logarithm, and so the eigenvalues of lie in the strip . When , the eigenvalues of , which are where is an eigenvalue of , lie in the segment of the complex plane.
The most important special case is for a positive integer , in which case
We can check that
so the definition does indeed produce a th root. The matrix is called the principal th root.
Returning to the case of rational , we note that
but can be a different matrix. In general, it is not true that for real and , although for symmetric positive definite matrices this identity does hold because the eigenvalues are real and positive.
An integral expression for valid for is
Another representation for with is given by the binomial expansion
For real matrices of the form
with there is an explicit formula for . It is easy to see that has eigenvalues , where . Let and . It can be shown that
The formula can be used computationally, but it is somewhat indirect in that one must approximate both the exponential and the logarithm. A more direct algorithm based on the Schur decomposition and Padé approximation of the power function is developed by Higham and Lin (2013). MATLAB code is available from MathWorks File Exchange.
If is diagonalizable, so that for some nonsingular with , then . This formula is safe to use computationally only if is well conditioned. For defective (non-diagonalizable) we can express in terms of the Jordan canonical form, but this expression is not useful computationally because the Jordan canonical form cannot be reliably computed.
If then , by the definition of square root. If does it follow that ? Clearly, the answer is “no” in general because, for example, does not imply .
Using the matrix unwinding function it can be shown that for . Hence the function is the inverse function of for .
How can we check the quality of an approximation to ? For we can check the residual , but for real there is no natural residual. Instead we can look at the backward error.
For an approximation to , a backward error is a matrix such that . Assume that and are nonsingular and that . Then, as shown in the previous section, implies . Hence the normwise relative backward error is
Applications with Stochastic Matrices
An important application of fractional matrix powers is in discrete-time Markov chains, which arise in areas including finance and medicine. A transition matrix for a Markov process is a matrix whose element is the probability of moving from state to state over a time step. It has nonnegative entries and the rows sum to , so it is a stochastic matrix. In practice, a transition matrix may be estimated for a certain time period, say one year, but a transition matrix for a shorter period, say one month, may be needed. If is a transition matrix for a time period then a stochastic th root of is a transition matrix for a time period a factor smaller. Therefore (years to months) and (weeks to days) are among the values of interest. Unfortunately, is not necessarily a stochastic matrix. Moreover, can have a stochastic th root that is not . For example, the stochastic matrix
has principal square root
but is not stochastic because of the negative elements. The square root
is stochastic, though. (Interestingly, is also a square root of !)
A wide variety configurations is possible as regards existence, nature (primary or nonprimary), and number of stochastic roots. Higham and Lin (2011) delineate the various possibilities that can arise. They note that the stochastic lower triangular matrix
has a stochastic th root, namely , for all . For example, to three significant figures,
The existence of stochastic roots of stochastic matrices is connected with the embeddability problem, which asks when a nonsingular stochastic matrix can be written for some with for and , . Kingman showed in 1962 that this condition holds if and only if for every positive integer there exists a stochastic such that .
Applications in Fractional Differential Equations
Fractional matrix powers arise in the numerical solution of differential equations of fractional order, especially partial differential equations involving fractional Laplace operators. Here, the problem may be one of computing , in which case for large problems it is preferable to directly approximate , for example by Krylov methods or quadrature methods, rather than to explicitly compute .
This is a minimal set of references, which contain further useful references within.
- Brian Davies, Embeddable Markov Matrices, Electronic J. Probability 15, 1474–1486, 2010.
- Roberto Garrappa and Marina Popolizio, On the Use of Matrix Functions for Fractional Partial Differential Equations, Math. Comput. Simulation C-25(81), 1045–1056, 2011.
- Nicholas J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
- Nicholas J. Higham and Lijing Lin, On th Roots of Stochastic Matrices, Linear Algebra Appl. 435(3), 448–463, 2011.
- Nicholas Higham and Lijing Lin, An Improved Schur–Padé Algorithm for Fractional Powers of a Matrix and their Fréchet Derivatives, SIAM J. Matrix Anal. Appl. 34(3), 1341–1360, 2013.
- Emmanuel Lorin and Simon Tian, A Numerical Study of Fractional Linear Algebraic Systems, Math. Comput. Simulation 182, 495–513, 2021.
Related Blog Posts
- Update of Catalogue of Software for Matrix Functions (2020)
- What Is a Matrix Function? (2020)
- What Is a Matrix Square Root? (2020)
- What Is the Matrix Exponential? (2020)
- What Is the Matrix Logarithm? (2020)
- What Is the Matrix Unwinding Function? (2020)
This article is part of the “What Is” series, available from https://nhigham.com/category/what-is and in PDF form from the GitHub repository https://github.com/higham/what-is.
3 thoughts on “What Is a Fractional Matrix Power?”
The fractional powers of unitary matrices are not unique because of the phase freedom. Does the integral representation take this fact into account? Are there any other constraints on fractional powers of unitary matrices?
A^\alpha as defined here is unique even for unitary matrices: the (principal) log in the definition takes care of phase freedom. So unitary matrices are not special as regards fractional powers.
Dear professor Nick, thank you for your reply. I have been trying to understand the fractional powers of unitaries. I know you had cited a few references in the post. It seems that these references mostly talk about these matrices in a more general sense. It would be nice to see if any other papers/articles that deal with unitary cases rigorously.