Matrix function

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.

Definitions

Most functions like ${\displaystyle \exp(x)}$ are defined as a solution of a differential equation. 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:

${\displaystyle \exp A:=\sum _{k=0}^{\infty }{\frac {1}{k!}}A^{k},}$

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

Importance

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

• Matrix exponential
• Root of a matrix
• Matrix cosine and sine
• Logarithm of a matrix
• Validated numerics for the functions above
• Matrix version of the gamma function

Related pages

• Matrix analysis
• Linear algebra
  • Numerical linear algebra

References

Further reading

• 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.