Lesson 18 - Eigenvalue Applications¶

Learning goals. By the end of this lecture, students should be able to:

recognize engineering problems where eigenvalues describe growth, decay, stability, or modal structure,
interpret dominant eigenvalues and eigenvectors in reactor-physics or transport-motivated examples,
use eigenvalue computations to analyze a simplified coupled system,
connect eigenmodes to transient behavior, stability, and long-term system response,
communicate what an eigenvalue result means physically rather than treating it as only an algebraic output.

Key terms. dominant eigenvalue, eigenmode, stability, transient response, long-term behavior.

What eigenvalues add to the story¶

The previous two chapters developed the mechanics of eigenvalue computation. Lesson 16 showed how repeated multiplication can reveal one dominant eigenpair. Lesson 17 showed how the characteristic polynomial and numerical eigensolvers describe the whole spectrum.

This chapter focuses on interpretation. In applications, eigenvalues are rarely the final answer by themselves. They become useful when we connect them to questions such as:

Does a process grow or decay?
Which mode dominates after many steps?
Is an iteration stable?
Which shape or pattern is associated with a multiplication factor?
How do coupled variables move together?

Takeaway.

An eigenvalue is a growth, decay, oscillation, or stability factor attached to a mode. The eigenvector tells us the shape of that mode.

Discrete-time growth and decay¶

Suppose a system evolves by repeated application of a matrix:

\[\mathbf{x}^{(k+1)}=\mathbf{G}\mathbf{x}^{(k)}.\]

If \(\mathbf{G}\mathbf{v}=\lambda\mathbf{v}\), then a state exactly in the \(\mathbf{v}\) direction evolves as

\[\mathbf{x}^{(k)}=\lambda^k\mathbf{v}.\]

Thus:

if \(|\lambda|<1\), that mode decays,
if \(|\lambda|>1\), that mode grows,
if \(\lambda<0\), the mode flips sign each step,
if \(\lambda\) is complex, the mode can oscillate while growing or decaying.

If \(\mathbf{G}\) has a full set of eigenvectors, a general initial condition can be decomposed into modes:

\[\mathbf{x}^{(0)}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_n\mathbf{v}_n.\]

After \(k\) steps,

\[\mathbf{x}^{(k)}=c_1\lambda_1^k\mathbf{v}_1+c_2\lambda_2^k\mathbf{v}_2+\cdots+c_n\lambda_n^k\mathbf{v}_n.\]

The largest-magnitude eigenvalues control the long-term behavior.

Stability of an iterative method¶

Stationary iterations from Lesson 14 can be written as

\[\mathbf{x}^{(k+1)}=\mathbf{G}\mathbf{x}^{(k)}+\mathbf{g}.\]

Let \(\mathbf{x}^{\ast}\) be the fixed point. The error

\[\mathbf{e}^{(k)}=\mathbf{x}^{(k)}-\mathbf{x}^{\ast}\]

satisfies

\[\mathbf{e}^{(k+1)}=\mathbf{G}\mathbf{e}^{(k)}.\]

Therefore, the iteration converges when every error mode decays. In eigenvalue language, the condition is

\[\rho(\mathbf{G})<1,\]

where \(\rho(\mathbf{G})\) is the spectral radius, the largest magnitude of any eigenvalue of \(\mathbf{G}\).

For source iteration in a simple scattering model,

\[\mathbf{G}=c\mathbf{P}.\]

When \(c\) is close to one, the dominant eigenvalue of \(c\mathbf{P}\) is close to one in magnitude, and convergence is slow. This connects the physical idea of long particle survival to the numerical idea of slow residual reduction.

A transport-motivated mode calculation¶

Consider a small nearest-neighbor transport operator

\[\begin{split}\mathbf{P}=\begin{bmatrix} 0 & 1/2 & 0 & 0\\ 1/2 & 0 & 1/2 & 0\\ 0 & 1/2 & 0 & 1/2\\ 0 & 0 & 1/2 & 0 \end{bmatrix}.\end{split}\]

This matrix maps a cell population to the population redistributed to neighboring cells. Boundary leakage is represented by the missing outward neighbors. With scattering ratio \(c\), the error-propagation matrix for source iteration is

\[\mathbf{G}=c\mathbf{P}.\]

The eigenvectors of \(\mathbf{P}\) are spatial modes. Smooth modes tend to decay slowly, while highly oscillatory modes often decay faster. The dominant eigenvalue of \(c\mathbf{P}\) predicts the slowest asymptotic convergence rate.

Listing. Eigenvalues of a small transport iteration matrix.

import numpy as np

P = np.array([[0.0, 0.5, 0.0, 0.0],
              [0.5, 0.0, 0.5, 0.0],
              [0.0, 0.5, 0.0, 0.5],
              [0.0, 0.0, 0.5, 0.0]])
c = 0.9
G = c * P
w, V = np.linalg.eig(G)
print(w)
print(max(abs(w)))

A residual history from source iteration should eventually decay at a rate close to the dominant magnitude \(\rho(\mathbf{G})\), assuming the initial error contains that mode.

Continuous-time systems¶

Eigenvalues also describe systems of differential equations. Consider

\[\frac{d\mathbf{x}}{dt}=\mathbf{A}\mathbf{x}.\]

If \(\mathbf{A}\mathbf{v}=\lambda\mathbf{v}\), then

\[\mathbf{x}(t)=e^{\lambda t}\mathbf{v}\]

is a solution. When \(\mathbf{A}\) is diagonalizable, its eigenvectors form a basis, so a general initial condition can be written as a sum of modal pieces. In that eigenvector basis, the coupled system decouples into scalar exponential factors. That is why the diagonalization condition from Lesson 17 is not a technical footnote; it is the bridge from matrix eigenvalues to modal ODE solutions such as point-kinetics-style systems.

Thus the real part of \(\lambda\) controls growth or decay:

\(\operatorname{Re}(\lambda)<0\): the mode decays,
\(\operatorname{Re}(\lambda)>0\): the mode grows,
\(\operatorname{Im}(\lambda)\neq 0\): the mode oscillates.

For example, let

\[\begin{split}\mathbf{A}=\begin{bmatrix} -0.8 & 0.3\\ 0.2 & -0.5 \end{bmatrix}.\end{split}\]

The diagonal terms represent loss from each component, while the off-diagonal terms represent coupling between components. The eigenvalues are approximately

\[\lambda_1\approx -0.9372, \qquad \lambda_2\approx -0.3628.\]

Both modes decay, but the second mode decays more slowly. Therefore, after the faster mode has died away, the state is dominated by the eigenvector associated with \(-0.3628\).

Coupled variables move as modes¶

An eigenvector tells us how coupled variables move together. For the continuous-time example, the slow eigenvector has entries with the same sign or opposite sign depending on the coupling. If two entries have the same sign, those variables rise and fall together in that mode. If they have opposite signs, one increases relative to the other.

This interpretation is often more important than the numerical value alone. A computed eigenvalue might say “decay rate \(-0.3628\),” but the associated eigenvector says which combination of variables decays at that rate.

A simplified two-group reactor-physics eigenproblem¶

A common reactor-physics eigenvalue problem can be written schematically as

\[\mathbf{M}\boldsymbol{\phi}=\frac{1}{k}\mathbf{F}\boldsymbol{\phi},\]

where \(\boldsymbol{\phi}\) is a flux vector, \(\mathbf{M}\) represents losses and transfers, \(\mathbf{F}\) represents fission production, and \(k\) is a multiplication factor. If \(\mathbf{M}\) is invertible, then

\[\mathbf{M}^{-1}\mathbf{F}\boldsymbol{\phi}=k\boldsymbol{\phi}.\]

The dominant eigenvalue is the multiplication factor, and the dominant eigenvector is the corresponding asymptotic flux shape.

As a toy two-group example, take

\[\begin{split}\mathbf{M}=\begin{bmatrix} 1.0 & -0.1\\ -0.4 & 0.7 \end{bmatrix}, \qquad \mathbf{F}=\begin{bmatrix} 0.8 & 0.5\\ 0 & 0 \end{bmatrix}.\end{split}\]

Then

\[\begin{split}\mathbf{B}=\mathbf{M}^{-1}\mathbf{F} \approx \begin{bmatrix} 0.8485 & 0.5303\\ 0.4848 & 0.3030 \end{bmatrix}.\end{split}\]

The eigenvalues of \(\mathbf{B}\) are approximately

\[k_1\approx 1.1515, \qquad k_2\approx 0.\]

Because the second eigenvalue is zero, the dominance ratio for this toy problem is

\[\left|\frac{k_2}{k_1}\right|=0.\]

That makes the example a deliberately clean one-step version of the first iteration demo: after one application, any component in the zero-eigenvalue mode is removed, and the normalized state is already aligned with the \(k_{\infty}\) flux shape. A normalized dominant eigenvector is approximately

\[\begin{split}\boldsymbol{\phi}\approx \begin{bmatrix} 0.8682\\ 0.4961 \end{bmatrix}.\end{split}\]

The interpretation is:

the multiplication factor is greater than one in this toy model, so production exceeds losses in the dominant mode,
the first group has the larger flux component in the dominant flux shape,
repeated source-production-loss application would align with this flux shape.

Listing. Solving a toy two-group eigenvalue problem.

import numpy as np

M = np.array([[1.0, -0.1],
              [-0.4, 0.7]])
F = np.array([[0.8, 0.5],
              [0.0, 0.0]])

B = np.linalg.solve(M, F)   # M^{-1} F without forming M^{-1}
w, V = np.linalg.eig(B)
idx = np.argmax(np.abs(w))
k = w[idx]
phi = V[:, idx]
phi = phi / np.linalg.norm(phi)
if phi[0] < 0:
    phi = -phi

print(k)
print(phi)
print(np.linalg.norm(B @ phi - k * phi))

The residual check at the end verifies the computed eigenpair.

Subcritical, critical, and supercritical interpretation¶

In reactor-physics language, a multiplication factor is often interpreted relative to one:

\(k<1\): subcritical, losses dominate production,
\(k=1\): critical, production balances losses,
\(k>1\): supercritical, production exceeds losses.

The eigenvector gives the shape associated with that balance. A statement such as \(k=1.15\) is incomplete unless we also understand the model assumptions, units, and flux shape represented by the eigenvector.

Full-spectrum versus dominant-mode thinking¶

The dominant eigenpair is often the first quantity of interest, but the full spectrum can answer different questions. In a transient problem, subdominant eigenvalues tell us how quickly non-dominant modes disappear. In an iterative method, the spread of eigenvalues influences convergence. In a coupled physical system, multiple eigenvectors can reveal multiple patterns of response.

The choice between one eigenpair and the full spectrum is therefore an engineering decision:

use one dominant eigenpair when the leading long-term mode is the main quantity of interest,
use several eigenpairs when transient or subdominant modes matter,
use the full spectrum for small dense systems when complete modal information is useful,
use sparse iterative eigensolvers for large systems when only a few modes are needed.

Communicating an eigenvalue result¶

A good eigenvalue interpretation should include four pieces:

the operator: what matrix or linear map was analyzed,
the eigenvalue: the growth, decay, stability, or multiplication factor,
the eigenvector: the associated shape or coupled pattern,
the implication: what the result means for the physical or numerical system.

This is especially important in applications because the same number can mean different things for different operators. An eigenvalue near one may mean slow convergence for an iteration matrix, near-critical multiplication for a reactor model, or long-lived behavior in a transient model.

For example, instead of saying only “the dominant eigenvalue is 0.97,” say: “For the source-iteration error operator, the dominant eigenvalue magnitude is about 0.97, so the slowest error mode decays by only about three percent per iteration. That explains the slow residual convergence.”

Exercises¶

A discrete-time model has an eigenvalue \(\lambda=-0.8\). What happens to that mode as the model is applied repeatedly?
An error-propagation matrix has spectral radius \(0.97\). What does that imply about asymptotic convergence?
A continuous-time linear model has an eigenvalue \(-0.5+2i\). Interpret the real and imaginary parts.
A toy reactor eigenproblem gives dominant multiplication factor \(k=1.15\) with a positive flux-shape eigenvector. What is the basic physical interpretation?

Answers.

The mode decays in magnitude because \(|\lambda|<1\), and it flips sign at each step because the eigenvalue is negative.
The slowest error mode decreases by only about three percent per iteration, so convergence is expected to be slow.
The real part \(-0.5\) gives exponential decay. The imaginary part \(2\) gives oscillation with angular frequency \(2\).
In the toy model, \(k>1\) means production exceeds losses, so the system is supercritical. The eigenvector gives the relative flux shape or component balance associated with that multiplication factor.

End-of-lecture summary¶