Eigenvalues/Eigenvectors of a $2\times2$ Matrix, and the Associated Linear IVP System: 04-BS-1, May 2014, Q4

Question 4: Eigenvalues/Eigenvectors of a $2\times2$ Matrix, and the Associated Linear IVP System (a) 8, (b) 12 marks

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Matrix $A=\begin{pmatrix}3&1\\-2&1\end{pmatrix}$; the linear system $\mathbf x'=A\mathbf x$ with $\mathbf x(0)=(1,0)$.

Find. (a) Eigenvalues and eigenvectors of $A$. (b) $x(t),y(t)$.

Approach. (a) Standard characteristic-polynomial eigenanalysis (the roots turn out complex). (b) Since the system's coefficient matrix is exactly $A$, build the real general solution from the complex eigenpair $\alpha\pm i\beta$ using $e^{\alpha t}[\mathbf u\cos\beta t\mp\mathbf v\sin\beta t]$, then fix the two constants from the ICs.

(a) Characteristic equation. $$\det(A-\lambda I)=(3-\lambda)(1-\lambda)+2=\lambda^2-4\lambda+5=0\ \Rightarrow\ \lambda=\frac{4\pm\sqrt{16-20}}{2}=2\pm i.$$
(a) Eigenvector for $\lambda=2+i$. $(A-\lambda I)\mathbf v=0$: row 1 gives $(1-i)v_1+v_2=0\Rightarrow v_2=(i-1)v_1$. Taking $v_1=1$: $$\boxed{\lambda=2+i:\ \mathbf v=(1,\ i-1)},\qquad \lambda=2-i:\ \mathbf v=(1,\ -i-1)\ \text{(conjugate)}.$$
(b) Real solution basis. Write $\mathbf v=\mathbf u+i\mathbf w$ with $\mathbf u=(1,-1)$, $\mathbf w=(0,1)$, $\alpha=2,\ \beta=1$. Two independent real solutions are $$\mathbf x_1=e^{2t}(\mathbf u\cos t-\mathbf w\sin t),\qquad \mathbf x_2=e^{2t}(\mathbf u\sin t+\mathbf w\cos t).$$ Componentwise: $x=e^{2t}(C_1\cos t+C_2\sin t)$, $y=e^{2t}[(-C_1+C_2)\cos t+(-C_1-C_2)\sin t]$.
(b) Apply initial conditions. $x(0)=C_1=1$. $y(0)=-C_1+C_2=0\Rightarrow C_2=1$. Then $y=e^{2t}[0\cdot\cos t-2\sin t]=-2e^{2t}\sin t$.

$$\boxed{x(t)=e^{2t}(\cos t+\sin t),\qquad y(t)=-2e^{2t}\sin t}$$

Quantity	Result
Eigenvalues	$2\pm i$
Eigenvector for $2+i$	$(1,\ i-1)$
$x(t)$	$e^{2t}(\cos t+\sin t)$
$y(t)$	$-2e^{2t}\sin t$

Concept & Theory

Topic. Complex eigenpairs and real solutions of a $2\times2$ linear ODE system.

Key relations. If $A$ has complex eigenpair $\lambda=\alpha\pm i\beta$ with eigenvector $\mathbf v=\mathbf u\pm i\mathbf w$, then $\mathbf x'=A\mathbf x$ has real solution basis $e^{\alpha t}(\mathbf u\cos\beta t-\mathbf w\sin\beta t)$ and $e^{\alpha t}(\mathbf u\sin\beta t+\mathbf w\cos\beta t)$.

Why this works. $\mathbf x(t)=e^{\lambda t}\mathbf v$ is a complex solution; writing $e^{(\alpha+i\beta)t}(\mathbf u+i\mathbf w)=e^{\alpha t}(\cos\beta t+i\sin\beta t)(\mathbf u+i\mathbf w)$ and separating real and imaginary parts (each itself a real solution, by linearity) gives exactly the two basis solutions above.

Common pitfall. Mixing up which eigenvector ($2+i$ vs $2-i$) supplies $\mathbf u,\mathbf w$, and sign slips differentiating the product of $e^{2t}$ with $\cos t,\sin t$ when checking the ODEs directly.

Source. Strang, Introduction to Linear Algebra, Ch. 6 (§6.3 Complex Eigenvalues, Systems of Differential Equations); Kreyszig, Advanced Engineering Mathematics, Ch. 4 (§4.3 Systems with Complex Eigenvalues).

Question 4 of 8: Eigenvalues/Eigenvectors of a $2\times2$ Matrix, and the Associated Linear IVP System

Question 4: Eigenvalues/Eigenvectors of a $2\times2$ Matrix, and the Associated Linear IVP System (a) 8, (b) 12 marks