Matrix Exponential

Definition

Given a square matrix $A$, the matrix exponential $e^{At}$ of $At$ is defined as the series:

$e^{At}={\sum }_{n=0}^{\infty }\frac{(At{)}^n}{n!}=I+At+\frac{(At{)}^2}{2!}+\frac{(At{)}^3}{3!}+\cdots$

where $I$ is the identity matrix.

Interpretation

The matrix exponential generalizes the scalar exponential $e^{at}$ to square matrices. It is the fundamental solution to the system of linear differential equations $\frac{d}{dt}P(t)=AP(t)$.

Key Property

If $A$ is diagonalizable as $A=M\Lambda M^{-1}$, where $\Lambda$ is diagonal matrix of eigenvalues and $M$ is the matrix of eigenvectors, then:

$e^{At}=M{e}^{\Lambda t}{M}^{-1}$

where $e^{\Lambda t}$ is the diagonal matrix with entries $e^{{\lambda }_{i}t}$.

This provides a computationally practical way to compute $e^{At}$.

Role in CTMCs

For a transition rate matrix $Q$, the transition probability matrix is:

$P(t)=e^{Qt}$

which solves both the forward and backward Kolmogorov equations.

Related

• Derivative of Matrix Exponential
• Transition Rate Matrix
• Kolmogorov Differential Equations

Exercises

Back to Roadmap 📖 → 🃏 → ✏

CTMC 2-state. Untuk $Q=\left[\begin{array}{cc}-\lambda & \lambda \\ \mu & -\mu \end{array}\right]$, hitung $P(t)=e^{Qt}$ menggunakan diagonalisasi.

Jawaban: Eigenvalues ${\lambda }_1=0$, ${\lambda }_2=-(\lambda +\mu )$. Eigenvectors $v_1=(\mu ,\lambda {)}^T$, $v_2=(1,-1{)}^T$. Maka: $P(t)=\frac{1}{\lambda +\mu }\left[\begin{array}{cc}\mu & \lambda \\ \mu & \lambda \end{array}\right]+\frac{e^{-(\lambda +\mu )t}}{\lambda +\mu }\left[\begin{array}{cc}\lambda & -\lambda \\ -\mu & \mu \end{array}\right]$

Verifikasi. Periksa bahwa ${\lim }_{t\to \infty }P(t)=\left[\begin{array}{cc}\frac{\mu }{\lambda +\mu }& \frac{\lambda }{\lambda +\mu }\\ \frac{\mu }{\lambda +\mu }& \frac{\lambda }{\lambda +\mu }\end{array}\right]$.

Baris identik → limit probabilitas independen dari state awal: ${\pi }_0=\frac{\mu }{\lambda +\mu }$, ${\pi }_1=\frac{\lambda }{\lambda +\mu }$.