What Is a Fractional Matrix Power?

A $p$ th root of an $n\times n$ matrix $A$ is a matrix $X$ such that $X^p = A$ , and it can be written $X = A^{1/p}$ . For a rational number $r = j/k$ (where $j$ and $k$ are integers), defining $A^r$ is more difficult: is it $(A^j)^{1/k}$ or $(A^{1/k})^j$ ? These two possibilities can be different even for $n = 1$ . More generally, how can we define $A^\alpha$ for an arbitrary real number $\alpha$ ?

Recall, first, that for a nonzero complex scalar $z$ we define $z^\alpha = \mathrm{e}^{\alpha\log z}$ , where $\log$ is the principal logarithm: the one taking values in the strip $\{\, z : -\pi < \mathop{\mathrm{Im}} z \le \pi \,\}$ . We can generalize this definition to matrices. For a nonsingular matrix $A$ we define

$\notag A^\alpha = \mathrm{e}^{\alpha\log A}.$

Here the logarithm is the principal matrix logarithm, the matrix function built on the principal scalar logarithm, and so the eigenvalues of $\log A$ lie in the strip $\{\, z : -\pi < \mathop{\mathrm{Im}} z \le \pi \,\}$ . When $\alpha \in [-1,1]$ , the eigenvalues of $A^\alpha$ , which are $\lambda^\alpha$ where $\lambda$ is an eigenvalue of $A$ , lie in the segment $\{\, z: -|\alpha|\pi < \arg z \le |\alpha|\pi\,\}$ of the complex plane.

The most important special case is $\alpha = 1/p$ for a positive integer $p$ , in which case

$\notag A^{1/p} = \mathrm{e}^{\frac{1}{p}\log A}.$

We can check that

$\notag \bigl(A^{1/p}\bigr)^p = \bigl(\mathrm{e}^{\frac{1}{p}\log A}\bigr)^p = \mathrm{e}^{\log A} = A,$

so the definition does indeed produce a $p$ th root. The matrix $A^{1/p}$ is called the principal $p$ th root.

Returning to the case of rational $p$ , we note that

$\notag A^{j/k} = \mathrm{e}^{\frac{j}{k}\log A} = \Bigl(\mathrm{e}^{\frac{1}{k}\log A}\Bigr)^j = (A^{1/k})^j,$

but $(A^{j})^{1/k}$ can be a different matrix. In general, it is not true that $(A^\alpha)^\beta = (A^\beta)^\alpha$ for real $\alpha$ and $\beta$ , although for symmetric positive definite matrices this identity does hold because the eigenvalues are real and positive.

An integral expression for $A^\alpha$ valid for $\alpha \in(0,1)$ is

$\notag A^\alpha = \displaystyle\frac{\sin(\alpha\pi)} {\alpha\pi} A \int_0^{\infty}(t^{1/\alpha}I+A)^{-1}\,\mathrm{d}t, \quad \alpha \in(0,1).$

Another representation for $A = t (I - B)$ with $\rho(B) < 1$ is given by the binomial expansion

$\notag A^{\alpha} = t^{\alpha} \displaystyle\sum_{j=0}^\infty {\alpha \choose j}(-B)^j, \quad \rho(B) < 1.$

For $2\times 2$ real matrices of the form

$\notag A = \begin{bmatrix} a & b \\ c & a \end{bmatrix}$

with $bc < 0$ there is an explicit formula for $A^\alpha$ . It is easy to see that $A$ has eigenvalues $\lambda_{\pm} = a \pm \mathrm{i} d$ , where $d = (-bc)^{1/2}$ . Let $\theta = \arg(\lambda_+) \in (0,\pi)$ and $r = |\lambda_+|$ . It can be shown that

$\notag A^\alpha = \displaystyle\frac{r^\alpha}{d} \begin{bmatrix} d\cos(\alpha\theta) & b\sin(\alpha\theta) \\ c\sin(\alpha\theta) & d\cos(\alpha\theta) \end{bmatrix}.$

Computation

The formula $A^\alpha = \mathrm{e}^{\alpha \log A}$ 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 $A$ is diagonalizable, so that $A = XDX^{-1}$ for some nonsingular $X$ with $D = \mathrm{diag}(\lambda_i)$ , then $A^\alpha = X D^\alpha X^{-1} = X \mathrm{diag}(\lambda_i^\alpha) X^{-1}$ . This formula is safe to use computationally only if $X$ is well conditioned. For defective (non-diagonalizable) $A$ we can express $A^\alpha$ in terms of the Jordan canonical form, but this expression is not useful computationally because the Jordan canonical form cannot be reliably computed.

Inverse Function

If $X = A^{1/2}$ then $A = X^2$ , by the definition of square root. If $X = A^{\alpha}$ does it follow that $A = X^{1/\alpha}$ ? Clearly, the answer is “no” in general because, for example, $X = A^2$ does not imply $A = X^{1/2}$ .

Using the matrix unwinding function it can be shown that $(A^{\alpha})^{1/\alpha} = A$ for $\alpha \in [-1,1]$ . Hence the function $g(A) = A^{1/\alpha}$ is the inverse function of $f(A) = A^{\alpha}$ for $\alpha\in[-1,1]$ .

Backward Error

How can we check the quality of an approximation $X$ to $A^{\alpha}$ ? For $\alpha = 1/p$ we can check the residual $A - X^p$ , but for real $p$ there is no natural residual. Instead we can look at the backward error.

For an approximation $X$ to $A^\alpha$ , a backward error is a matrix $\Delta A$ such that $X = (A + \Delta A)^\alpha$ . Assume that $A$ and $X$ are nonsingular and that $\alpha \in [-1,1]$ . Then, as shown in the previous section, $X = (A + \Delta A)^\alpha$ implies $\Delta A = X^{1/\alpha} - A$ . Hence the normwise relative backward error is

$\notag \eta(X) = \displaystyle\frac{ \|X^{1/\alpha} - A \|}{\|A\|}, \quad \alpha\in[-1,1].$

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 $(i,j)$ element is the probability of moving from state $i$ to state $j$ over a time step. It has nonnegative entries and the rows sum to $1$ , 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 $A$ is a transition matrix for a time period $p$ then a stochastic $p$ th root of $A$ is a transition matrix for a time period a factor $p$ smaller. Therefore $p = 12$ (years to months) and $p = 7$ (weeks to days) are among the values of interest. Unfortunately, $A^{1/p}$ is not necessarily a stochastic matrix. Moreover, $A$ can have a stochastic $p$ th root that is not $A^{1/p}$ . For example, the stochastic matrix

$\notag A = \left[\begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\\ \end{array} \right]$

has principal square root

$\notag A^{1/2} = \frac{1}{3} \left[\begin{array}{rrr} 2 & 2 & -1 \\ -1 & 2 & 2 \\ 2 & -1 & 2 \end{array} \right],$

but $A^{1/2}$ is not stochastic because of the negative elements. The square root

$\notag Y = \left[\begin{array}{ccc} 0 & 0 & 1\\ 1 & 0 & 0\\ 0 & 1 & 0\\ \end{array} \right]$

is stochastic, though. (Interestingly, $A$ is also a square root of $Y$ !)

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

$\notag A_n = \begin{bmatrix} 1 & & & \\ \frac{1}{2} & \frac{1}{2} & & \\ \vdots & \vdots &\ddots & \\ \frac{1}{n} & \frac{1}{n} &\cdots & \frac{1}{n} \end{bmatrix}$

has a stochastic $p$ th root, namely $A_n^{1/p}$ , for all $p$ . For example, to three significant figures,

$\notag A_6^{1/3} = \begin{bmatrix} 1.000 & & & & & \\ 0.206 & 0.794 & & & & \\ 0.106 & 0.201 & 0.693 & & & \\ 0.069 & 0.111 & 0.190 & 0.630 & & \\ 0.050 & 0.075 & 0.109 & 0.181 & 0.585 & \\ 0.039 & 0.056 & 0.076 & 0.107 & 0.172 & 0.550 \\ \end{bmatrix}.$

The existence of stochastic roots of stochastic matrices is connected with the embeddability problem, which asks when a nonsingular stochastic matrix $A$ can be written $A = \mathrm{e}^Q$ for some $Q$ with $q_{ij} \ge 0$ for $i\ne j$ and $\sum_{j=1}^n q_{ij} = 0$ , $i=1\colon n$ . Kingman showed in 1962 that this condition holds if and only if for every positive integer $p$ there exists a stochastic $X_p$ such that $A = X_p^p$ .

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 $A^\alpha b$ , in which case for large problems it is preferable to directly approximate $A^\alpha b$ , for example by Krylov methods or quadrature methods, rather than to explicitly compute $A^\alpha$ .

References

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 $p$ 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.

3 thoughts on “What Is a Fractional Matrix Power?”

Dileep Varikuti (@vndileep14n) says:

December 22, 2021 at 6:38 am

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?

1. Nick Higham says:
  
  December 22, 2021 at 7:34 am
  
  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.
  
  1. Dileep Varikuti (@vndileep14n) says:
    
    December 22, 2021 at 10:43 am
    
    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.

What Is a Fractional Matrix Power?

Computation

Inverse Function

Backward Error

Applications with Stochastic Matrices

Applications in Fractional Differential Equations

References

Related Blog Posts

3 thoughts on “What Is a Fractional Matrix Power?”

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Computation

Inverse Function

Backward Error

Applications with Stochastic Matrices

Applications in Fractional Differential Equations

References

Related Blog Posts

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3 thoughts on “What Is a Fractional Matrix Power?”

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