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.^[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:

• Matrix exponential^{[7][8][9][10][11]}
• Root of a matrix^[12]
• Matrix cosine and sine^[13]
• Logarithm of a matrix^[14]
• Validated numerics for the functions above^[15][16][17]
• Matrix version of the gamma function^[18]

Related pages

• Matrix analysis
• Linear algebra
  • Numerical linear algebra