The Eigenvalue Method

This chapter reveals why eigenvalues and eigenvectors matter so profoundly. They are not just abstract concepts from linear algebra. They are the key to understanding how dynamical systems behave, predicting whether solutions grow or decay, and writing down explicit formulas for trajectories in phase space.

The Central Question

We have a linear system:

where $A$ is a constant $2 \times 2$ matrix. We want to find all solutions. But how?

For a scalar equation $x' = ax$, the solution is simply $x(t) = x_0 e^{at}$. The exponential function solves it because differentiation pulls down a factor of $a$: $(e^{at})' = ae^{at}$.

Can we find something similar for systems? The answer is yes, and the key lies in finding special directions where the matrix $A$ acts like a simple scalar multiplication.

Eigenvectors: The Special Directions

An eigenvector $\mathbf{v}$ of matrix $A$ is a nonzero vector that is simply scaled when multiplied by $A$:

The scalar $\lambda$ is called the eigenvalue associated with $\mathbf{v}$. In this special direction, the matrix $A$ acts like multiplying by the number $\lambda$.

Here is the crucial insight: if we can find an eigenvector of $A$, we can construct a solution to $\mathbf{x}' = A\mathbf{x}$.

Why Eigenvectors Give Solutions

Suppose $\mathbf{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$. Consider the function:

Let us verify this is a solution. The left side of $\mathbf{x}' = A\mathbf{x}$ is:

The right side is:

Both sides are equal. We have found a solution.

This is the essence of the eigenvalue method: eigenvectors point in directions where motion stays along that direction. A solution starting on an eigenvector line simply grows or decays exponentially along that line, depending on the sign of $\lambda$.

Notice how trajectories that start along an eigenvector direction remain on that line forever. They simply scale: growing if $\lambda > 0$, decaying if $\lambda < 0$. These are the simplest possible solutions to the system.

Finding Eigenvalues

To find eigenvalues, we start from $A\mathbf{v} = \lambda \mathbf{v}$, which can be rewritten as:

For this to have a nonzero solution $\mathbf{v}$, the matrix $(A - \lambda I)$ must be singular. This happens exactly when its determinant is zero:

This is the characteristic equation. For a $2 \times 2$ matrix:

the characteristic equation becomes:

Expanding:

This is a quadratic in $\lambda$. The solutions are the eigenvalues.

Finding Eigenvectors

Once we have an eigenvalue $\lambda$, we find its eigenvector by solving:

This gives a system of linear equations. Since the matrix is singular, the equations are dependent, and we get a family of solutions forming a line through the origin. We typically pick one nonzero vector from this line as our eigenvector.

The General Solution: Distinct Real Eigenvalues

If $A$ has two distinct real eigenvalues $\lambda_1$ and $\lambda_2$ with corresponding eigenvectors $\mathbf{v}_1$ and $\mathbf{v}_2$, then:

are two independent solutions. The general solution is their linear combination:

The constants $c_1$ and $c_2$ are determined by the initial condition $\mathbf{x}(0)$.

The general solution is a superposition of the two eigenvector solutions. Each component evolves independently: the component along $\mathbf{v}_1$ grows or decays at rate $\lambda_1$, while the component along $\mathbf{v}_2$ grows or decays at rate $\lambda_2$.

Applying Initial Conditions

Given an initial condition $\mathbf{x}(0) = \mathbf{x}_0$, we need to find $c_1$ and $c_2$ such that:

This is just expressing $\mathbf{x}_0$ as a linear combination of the eigenvectors. Since the eigenvectors form a basis (they are linearly independent), there is exactly one such decomposition.

Example: Consider $\mathbf{x}' = A\mathbf{x}$ where:

The characteristic equation is:

So $\lambda_1 = 3$ and $\lambda_2 = -1$.

For $\lambda_1 = 3$: Solving $(A - 3I)\mathbf{v} = \mathbf{0}$ gives $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$.

For $\lambda_2 = -1$: Solving $(A + I)\mathbf{v} = \mathbf{0}$ gives $\mathbf{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$.

The general solution is:

If $\mathbf{x}(0) = \begin{bmatrix} 2 \\ 0 \end{bmatrix}$, then $c_1 + c_2 = 2$ and $c_1 - c_2 = 0$, giving $c_1 = c_2 = 1$.

Complex Eigenvalues: Spiraling Solutions

The discriminant of the characteristic equation may be negative, giving complex eigenvalues. For real matrices, complex eigenvalues always come in conjugate pairs:

where $\alpha$ and $\beta$ are real. The real part $\alpha$ controls growth or decay. The imaginary part $\beta$ controls the rotation frequency.

Deriving the real solutions: The complex eigenvalue $\lambda = \alpha + \beta i$ has a complex eigenvector $\mathbf{v}$. The complex solution $e^{\lambda t}\mathbf{v} = e^{\alpha t}e^{i\beta t}\mathbf{v}$ involves complex exponentials. Using Euler's formula $e^{i\beta t} = \cos\beta t + i\sin\beta t$, we can write:

Separating this into real and imaginary parts gives two real, linearly independent solutions. If $\mathbf{v} = \mathbf{p} + i\mathbf{q}$ where $\mathbf{p}$ and $\mathbf{q}$ are real vectors, the two real solutions are:

The general solution is $\mathbf{x}(t) = c_1\mathbf{x}_1(t) + c_2\mathbf{x}_2(t)$.

The key geometric insight is simpler: solutions spiral around the origin.

The real part determines the fate of the spiral:

• $\alpha < 0$: stable spiral (solutions spiral inward toward the origin)
• $\alpha > 0$: unstable spiral (solutions spiral outward to infinity)
• $\alpha = 0$: center (solutions orbit in closed curves, neither approaching nor fleeing)

The imaginary part $\beta$ determines how fast the spiral rotates.

Repeated Eigenvalues

When the characteristic equation has a repeated root $\lambda$ (discriminant equals zero), we need a second, independent solution. The approach depends on whether there are one or two independent eigenvectors.

If there is only one eigenvector $\mathbf{v}$, the general solution takes the form:

where $\mathbf{w}$ is a generalized eigenvector satisfying $(A - \lambda I)\mathbf{w} = \mathbf{v}$.

The factor of $t$ appears, just as it did for repeated roots in second-order equations. This is the deficient case, producing a degenerate node.

The Complete Picture: From Matrix to Phase Portrait

The eigenvalues of $A$ completely determine the qualitative behavior of the phase portrait:

Adjust the matrix entries and watch how the eigenvalues change, and with them the entire character of the phase portrait. The connection between algebra (eigenvalues) and geometry (trajectories) is direct and powerful.

The Eigenvalue Method: Summary

The complete algorithm for solving $\mathbf{x}' = A\mathbf{x}$:

1. Find eigenvalues: Solve $\det(A - \lambda I) = 0$
2. Find eigenvectors: For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = \mathbf{0}$
3. Write general solution:
   • Distinct real: $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$
   • Complex $\alpha \pm \beta i$: Use $e^{\alpha t}\cos\beta t$ and $e^{\alpha t}\sin\beta t$
   • Repeated: Include a factor of $t$ and use generalized eigenvector if needed
4. Apply initial conditions: Solve for $c_1$ and $c_2$ from $\mathbf{x}(0) = \mathbf{x}_0$

Why This Matters

The eigenvalue method is more than a technique for solving equations. It reveals why linear systems behave the way they do.

Eigenvalues tell you about stability: are solutions attracted to an equilibrium or repelled from it? Eigenvectors tell you about geometry: in which directions does the attraction or repulsion occur?

This framework extends far beyond $2 \times 2$ systems. In higher dimensions, the same principles apply: eigenvalues classify behavior, and eigenvectors identify the special directions where that behavior is most apparent.

The next chapter will classify all possible behaviors systematically, connecting the eigenvalue structure to named patterns like nodes, spirals, saddles, and centers.

Key Takeaways

• If $A\mathbf{v} = \lambda \mathbf{v}$, then $\mathbf{x}(t) = e^{\lambda t}\mathbf{v}$ solves $\mathbf{x}' = A\mathbf{x}$
• Eigenvectors define directions where solutions move in straight lines, simply scaling by $e^{\lambda t}$
• Eigenvalues are found from the characteristic equation $\det(A - \lambda I) = 0$
• For distinct real eigenvalues, the general solution is $c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$
• Complex eigenvalues $\alpha \pm \beta i$ produce spiraling solutions; the real part controls decay/growth, the imaginary part controls rotation
• The sign of eigenvalues determines stability: negative means decay toward equilibrium, positive means growth away from it
• Initial conditions determine the coefficients $c_1$ and $c_2$ by decomposing the initial state into eigenvector components