Matrix Pseudospectrum

The pseudospectrum of a matrix generalizes the concept of eigenvalues, revealing how sensitive the spectrum is to small perturbations. For a nonnormal matrix, the region where the matrix nearly becomes singular can be much larger than the eigenvalues themselves.

Concept Overview

For a normal matrix (satisfying A^*A = AA^*), the resolvent norm ||(zI - A)^-1||₂ equals the reciprocal of the distance from z to the nearest eigenvalue, so the closed ε-pseudospectrum is simply the union of closed disks of radius ε around the eigenvalues.

For a nonnormal matrix such as A = [[λ, k], [0, λ]] with coupling k > 0, the eigenvectors collapse and the resolvent norm can be large far outside the ε-disk around λ.

Mathematical Formulation

The closed ε-pseudospectrum Λ_ε(A) is the set of complex numbers z where the smallest singular value satisfies σ_min(zI - A) ≤ ε (equivalently, ||(zI - A)^-1||₂ ≥ 1/ε, with ||(zI - A)^-1||₂ = ∞ at eigenvalues).

For the 2×2 upper-triangular block A = [[λ, k], [0, λ]] and radial distance squared r² = |z - λ|², the smallest singular value has the numerically stable closed form σ_min(zI - A) = 2r² / (√(k² + 4r²) + k), and Λ_ε(A) is an exact disk centered at λ of radius √(ε(ε + k)).

Worked Example

Consider the matrix A = [[0, 2], [0, 0]]. The eigenvalue is λ = 0. We can probe the complex plane at z = x + iy. Let's pick z = 1 and an ε = 0.5. The resolvent norm is given by the inverse of the smallest singular value of zI - A. Here, zI - A = [[1, -2], [0, 1]].

Using the formula for σ_min with r² = 1, k = 2: σ_min = (2 × 1) / (√(2² + 4 × 1) + 2) = 2 / (√8 + 2) ≈ 0.41421356237309503.
The resolvent norm is 1 / 0.41421356237309503 ≈ 2.414213562373095.
So we expect the calculated Smallest singular value to be 0.41421356237309503.
Since 0.41421356237309503 ≤ 0.5, this point lies within the closed ε-pseudospectrum. The exact pseudospectral radius for a 2x2 nilpotent block is √(ε(ε + k)), which is √(0.5 × 2.5) ≈ 1.118033988749895.

Experiment

Explore the exact pseudospectrum by adjusting the eigenvalue, nonnormality, and complex probe position. You can test different values to see if the probe falls inside the calculated ε radius disk.

Set Lambda to 0.
Set Coupling to 2.
Set Probe real to 1 and Probe imaginary to 0.
Set Epsilon to 0.5.
Observe that the probe turns green and the Within closed epsilon set shows true.

For more interactive exploration, open the full module.

Limitations

This visualization simplifies the matrix to a specific 2x2 Jordan-type block form where the pseudospectrum is an exact circle. In general, the pseudospectra of arbitrary higher-dimensional matrices have non-circular boundary curves and multiple connected components. Also, at exact eigenvalues (z = λ), the inverse (zI - A)^-1 does not exist and its norm is infinite by convention.

Common Pitfalls

Do not assume that the ε-pseudospectrum of a matrix with repeated eigenvalue λ has radius ε. When the nonnormal coupling k > 0 is large compared to ε, the pseudospectral radius √(ε(ε + k)) scales like √(kε), which is much larger than ε for small perturbations.

References

Pseudospectra Gateway

The Pseudospectra Gateway, maintained by L. N. Trefethen and M. Embree, provides a comprehensive introduction to the theory and applications of pseudospectra for nonnormal matrices and operators.

Matrix Pseudospectrum