Matrix Exponential

The matrix exponential is a function on square matrices analogous to the exponential function for real numbers. It is used to solve systems of linear differential equations and has applications in various fields such as control theory, quantum mechanics, and more.

Definition

For a square matrix $A$, the matrix exponential, denoted as $e^A$, is defined by the following power series:

$$e^A=\sum _{k=0}^{\infty }\frac{A^k}{k!}=I+A+\frac{A^2}{2!}+\frac{A^3}{3!}+\cdots$$

where $A^0$is the identity matrix $I$.

Properties

Elementary properties

1. Identity Matrix: If $A$ is the identity matrix $I$, then:

   $$e^I=e\cdot I$$

2. Diagonalizable Matrices: If $A$ is diagonalizable, i.e., $A=PD{P}^{-1}$where $D$ is a diagonal matrix, then:

   $$e^A=P{e}^D{P}^{-1}$$

   where $e^D$is the exponential of the diagonal matrix $D$, computed by exponentiating each diagonal element.

3. Nilpotent Matrices: If $A$ is nilpotent (i.e., there exists some $k$such that $A^k=0$), the series terminates after a finite number of terms.

Determinant

For any complex square matrix the following trace holds

$$\det (e^A)=e^{\mathrm{tr}(A)}$$

Trace

let