# Matrix function

function that maps matrices to matrices

In mathematics, a function maps an input value to an output value. In the case of a matrix function, the input and the output values are matrices. One example of a matrix function occurs with the Algebraic Riccati equation, which is used to solve certain optimal control problems.

Matrix functions are special functions made by matrices.[1]

## Definitions

Most functions like ${\displaystyle \exp(x)}$  are defined as a solution of a differential equation.[2] But matrix functions will use a different way.

Suppose ${\displaystyle z}$  is a complex number and ${\displaystyle A}$  is a square matrix. If you have a polynomial:

${\displaystyle f(z):=c_{0}+c_{1}z+\cdots +c_{m}z^{m}}$ ,

then it is reasonable to define

${\displaystyle f(A):=c_{0}I+c_{1}A+\cdots +c_{m}A^{m}.}$

Let's use this idea. When you have

${\displaystyle f(z):=\sum _{k=0}^{\infty }c_{k}z^{k}}$ ,

then you can introduce

${\displaystyle f(A):=\sum _{k=0}^{\infty }c_{k}A^{k}.}$

For example, the matrix version of the exponential function and the trigonometric functions are defined as follows:[1]

${\displaystyle \exp A:=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k},}$
${\displaystyle \sin A:=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)!}}A^{2k+1},\quad \cos A:=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!}}A^{2k}.}$

## Importance

Matrix functions are used at numerical methods for ordinary differential equations[3][4][5] and statistics.[1][6] This is why numerical analysts are studying how to compute them.[1] For example, the following functions are studied:

## References

1. Higham, Nicholas J. (2008). Functions of matrices theory and computation. Philadelphia: Society for Industrial and Applied Mathematics.
2. Andrews, G. E., Askey, R., & Roy, R. (1999). Special functions (Vol. 71). Cambridge University Press.
3. Hochbruck, M., & Ostermann, A. (2010). Exponential integrators. Acta Numerica, 19, 209-286.
4. Al-Mohy, A. H., & Higham, N. J. (2011). Computing the action of the matrix exponential, with an application to exponential integrators. SIAM journal on scientific computing, 33(2), 488-511.
5. Del Buono, N., & Lopez, L. (2003, June). A survey on methods for computing matrix exponentials in numerical schemes for ODEs. In International Conference on Computational Science (pp. 111-120). Springer, Berlin, Heidelberg.
6. James, A. T. (1975). Special functions of matrix and single argument in statistics. In Theory and Application of Special Functions (pp. 497-520). Academic Press.
7. Moler, C., & Van Loan, C. (1978). Nineteen dubious ways to compute the exponential of a matrix. SIAM review, 20(4), 801-836.
8. Moler, C., & Van Loan, C. (2003). Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM review, 45(1), 3-49.
9. Higham, N. J. (2005). The scaling and squaring method for the matrix exponential revisited. SIAM Journal on Matrix Analysis and Applications, 26(4), 1179-1193.
10. Sidje, R. B. (1998). Expokit: A software package for computing matrix exponentials. ACM Transactions on Mathematical Software (TOMS), 24(1), 130-156.
11. Yuka Hashimoto,Takashi Nodera, Double-shift-invert Arnoldi method for computing the matrix exponential, Japan J. Indust. Appl. Math, pp727-738, 2018.
12. Bini, D. A., Higham, N. J., & Meini, B. (2005). Algorithms for the matrix pth root. Numerical Algorithms, 39(4), 349-378.
13. Hargreaves, G. I., & Higham, N. J. (2005). Efficient algorithms for the matrix cosine and sine. Numerical Algorithms, 40(4), 383-400.
14. Hale, N., Higham, N. J., & Trefethen, L. N. (2008). Computing ${\displaystyle A^{\alpha },\log(A)}$ , and related matrix functions by contour integrals. SIAM Journal on Numerical Analysis, 46(5), 2505-2523.
15. Miyajima, S. (2019). Verified computation of the matrix exponential. Advances in Computational Mathematics, 45(1), 137-152.
16. Miyajima, S. (2019). Verified computation for the matrix principal logarithm. Linear Algebra and its Applications, 569, 38-61.
17. Miyajima, S. (2018). Fast verified computation for the matrix principal pth root. Journal of Computational and Applied Mathematics, 330, 276-288.
18. Joao R. Cardoso, Amir Sadeghi, Computation of matrix gamma function, BIT Numerical Mathematics, (2019)
• A Survey of the Matrix Exponential Formulae with Some Applications (2016), Baoying Zheng, Lin Zhang, Minhyung Cho, and Junde Wu. J. Math. Study Vol. 49, No. 4, pp. 393-428.
• Higham, N. J. (2006). Functions of matrices. Manchester Institute for Mathematical Sciences, School of Mathematics, The University of Manchester.