Distinct real eigenvalues case

For a constant-coefficient linear system ${\mathbf{x}}^{\prime }=A\mathbf{x}$ where $A$ is $n\times n$ with distinct real eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$, the general solution is a linear combination of exponentials:

$\mathbf{x}(t)=c_1{\mathbf{u}}_1{e}^{{\lambda }_{1}t}+c_2{\mathbf{u}}_2{e}^{{\lambda }_{2}t}+\cdots +c_n{\mathbf{u}}_n{e}^{{\lambda }_{n}t}$

where ${\mathbf{u}}_i$ is an eigenvector corresponding to ${\lambda }_i$.

This is the cleanest case. Distinct eigenvalues automatically give $n$ linearly independent eigenvectors, hence $n$ linearly independent solutions, hence a complete general solution.

Why eigenvectors give solutions

Plug $\mathbf{x}(t)=\mathbf{u}{e}^{\lambda t}$ into ${\mathbf{x}}^{\prime }=A\mathbf{x}$:

$\lambda \mathbf{u}{e}^{\lambda t}=A\mathbf{u}{e}^{\lambda t}$

Cancel $e^{\lambda t}$:

$A\mathbf{u}=\lambda \mathbf{u}$

So $\lambda$ must be an eigenvalue of $A$ and $\mathbf{u}$ a corresponding eigenvector. Lemma: $\mathbf{x}=\mathbf{u}{e}^{\lambda t}$ is a solution of ${\mathbf{x}}^{\prime }=A\mathbf{x}$ if and only if $(\lambda ,\mathbf{u})$ is an eigenvalue/eigenvector pair of $A$.

The procedure

1. Find eigenvalues: solve $\det (A-\lambda I)=0$.
2. For each eigenvalue ${\lambda }_i$, find an eigenvector: solve $(A-{\lambda }_{i}I){\mathbf{u}}_i=0$.
3. Write general solution: $\mathbf{x}(t)=\sum c_i{\mathbf{u}}_i{e}^{{\lambda }_{i}t}$.
4. Apply initial conditions to find the $c_i$.

Worked example

Solve ${\mathbf{x}}^{\prime }=\left[\begin{array}{cc}1& 1\\ 4& 1\end{array}\right]\mathbf{x}$.

Eigenvalues: $\det \left[\begin{array}{cc}1-\lambda & 1\\ 4& 1-\lambda \end{array}\right]=(1-\lambda {)}^2-4={\lambda }^2-2\lambda -3=0$.

Roots: ${\lambda }_1=-1$, ${\lambda }_2=3$. Distinct real.

Eigenvector for ${\lambda }_1=-1$: solve $(A+I){\mathbf{u}}_1=0$, i.e., $\left[\begin{array}{cc}2& 1\\ 4& 2\end{array}\right]{\mathbf{u}}_1=0$.

The two rows are dependent (second is twice the first), giving $2u_{1,1}+u_{1,2}=0$. Pick $u_{1,1}=1$, $u_{1,2}=-2$:

${\mathbf{u}}_1=\left[\begin{array}{c}1\\ -2\end{array}\right]$

Eigenvector for ${\lambda }_2=3$: solve $(A-3I){\mathbf{u}}_2=0$, i.e., $\left[\begin{array}{cc}-2& 1\\ 4& -2\end{array}\right]{\mathbf{u}}_2=0$.

Gives $-2u_{2,1}+u_{2,2}=0$. Pick $u_{2,1}=1$, $u_{2,2}=2$:

${\mathbf{u}}_2=\left[\begin{array}{c}1\\ 2\end{array}\right]$

General solution:

$\mathbf{x}(t)=c_1\left[\begin{array}{c}1\\ -2\end{array}\right]e^{-t}+c_2\left[\begin{array}{c}1\\ 2\end{array}\right]e^{3t}$

In components:

$\{\begin{array}{l}{x}_1(t)=c_1{e}^{-t}+c_2{e}^{3t}\\ x_2(t)=-2c_1{e}^{-t}+2c_2{e}^{3t}\end{array}$

Diagonal special case

If $A$ is diagonal, the eigenvalues are the diagonal entries and the eigenvectors are the standard basis vectors. The system decouples completely:

$A=\left[\begin{array}{ccc}{\lambda }_1& 0& 0\\ 0& {\lambda }_2& 0\\ 0& 0& {\lambda }_3\end{array}\right]\Rightarrow \{\begin{array}{l}{x}_1^{\prime }={\lambda }_1{x}_1\\ x_2^{\prime }={\lambda }_2{x}_2\\ x_3^{\prime }={\lambda }_3{x}_3\end{array}$

Each component is a 1D ODE: $x_i=c_i{e}^{{\lambda }_{i}t}$. The general solution is

$\mathbf{x}(t)=c_1{e}^{{\lambda }_{1}t}\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]+c_2{e}^{{\lambda }_{2}t}\left[\begin{array}{c}0\\ 1\\ 0\end{array}\right]+c_3{e}^{{\lambda }_{3}t}\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$

When $A$ isn’t diagonal but has distinct eigenvalues, the eigenvector basis effectively diagonalizes the system in a rotated frame — same idea, different coordinates.

Stability and qualitative behavior

Long-term behavior depends on the signs of the eigenvalues:

• All ${\lambda }_i<0$: solutions decay to zero. Origin is asymptotically stable (a stable node in 2D).
• All ${\lambda }_i>0$: solutions grow exponentially. Origin is unstable.
• Mixed signs: solutions in some directions decay, others grow. Origin is a saddle point (unstable).

Stable node: distinct negative real eigenvalues. Trajectories approach the origin tangent to the slower eigendirection.

For 2D phase portraits, see Phase plane behaviour cases 1, 2, 3.

For other eigenvalue scenarios, see Complex conjugate eigenvalues case and Repeated eigenvalues case.