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# Lecture 26 ## Topic - Matrix exponential - First order systems - Phase portraits ## Matrix Exponential ### Definition For a square matrix $A$, the matrix exponential is defined by $$ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} $$ ### P...

# Lecture 26 ## Topic - Matrix exponential - First order systems - Phase portraits ## Matrix Exponential ### Definition For a square matrix $A$, the matrix exponential is defined by $$ e^{At} = I + At + \frac{(At)^2}{2!} + \frac{(At)^3}{3!} + \cdots = \sum_{k=0}^{\infty} \frac{(At)^k}{k!} $$ ### Properties 1. $\frac{d}{dt} e^{At} = A e^{At} = e^{At} A$ 2. $(e^{At})^{-1} = e^{-At}$ 3. $e^{A(t+s)} = e^{At} e^{As}$ ### How to compute $e^{At}$ 1. If $A$ is diagonalizable, i.e. $A = PDP^{-1}$ where $D$ is diagonal, then $e^{At} = P e^{Dt} P^{-1}$ and $e^{Dt}$ is a diagonal matrix with diagonal entries $e^{\lambda_i t}$ where $\lambda_i$ are the eigenvalues of $A$. 2. If $(\lambda - \lambda_i)^k$ is a factor of the characteristic polynomial, then $e^{\lambda_i t}, te^{\lambda_i t}, \cdots, t^{k-1} e^{\lambda_i t}$ are factors of elements of $e^{At}$. Use initial conditions to solve for the coefficients. ## First Order Systems ### Definition $$ \frac{d\mathbf{x}}{dt} = A \mathbf{x} $$ where $\mathbf{x} \in \mathbb{R}^n$ and $A$ is an $n \times n$ matrix ### Solution $$ \mathbf{x}(t) = e^{At} \mathbf{x}(0) $$ ## Phase Portraits ### 2x2 Systems Consider $\frac{d\mathbf{x}}{dt} = A \mathbf{x}$ where $\mathbf{x} \in \mathbb{R}^2$ and $A$ is a $2 \times 2$ matrix 1. **Node:** Eigenvalues are real and have the same sign. * Stable Node: Eigenvalues are negative. * Unstable Node: Eigenvalues are positive. 2. **Saddle:** Eigenvalues are real and have opposite signs. Always unstable. 3. **Spiral:** Eigenvalues are complex with non-zero real part. * Stable Spiral: Eigenvalues have negative real part. * Unstable Spiral: Eigenvalues have positive real part. 4. **Center:** Eigenvalues are purely imaginary. Trajectories are closed curves (ellipses).