Recipe

Assess how much a residual guarantees about an eigenvalue

Use when: you have a candidate eigenpair (x,\lambda) with residual r=Ax-\lambda x and want to know how far \lambda actually sits from a genuine eigenvalue

Goal: a residual-based bound on the error when A is normal, or an honest admission that no such bound exists when A is not

Taught in Approximate eigenvectors and why the spectrum can be unstable

1. 1

   Multiply A by your unit-length candidate eigenvector x, then subtract \lambda x, to form r=Ax-\lambda x.

   Example

   For A=\begin{bmatrix}4&0\\0&2\end{bmatrix}, unit candidate x=(1,0), and \lambda=4.1: Ax=(4,0), so r=(4,0)-4.1(1,0)=(-0.1,0), giving \lVert r\rVert=0.1.

   Why: r is the only error signal available once the true eigenvalue is unknown, so every later step in this recipe works from this one number.

2. 2

   Test whether A commutes with its own conjugate transpose, i.e. whether AA^{*}=A^{*}A. A symmetric, Hermitian, orthogonal, or unitary matrix always passes this test.

   • AA^{*}=A^{*}A (A is normal) — continue to Step 3 for a guaranteed bound.
   • AA^{*}\ne A^{*}A (A is not normal) — skip to Step 4; Step 3's bound no longer applies.

   Why: only a normal matrix has an orthonormal eigenbasis, and that orthonormality is exactly what Step 3's bound depends on.

3. 3

   Conclude that the smallest gap between \lambda and any genuine eigenvalue of A satisfies \min_i|\lambda_i-\lambda|\le\lVert r\rVert.

   Example

   Continuing the Step 1 example, the residual norm is 0.1, and A's true eigenvalues are 4 and 2. The nearest one, 4, sits exactly 0.1 away, matching the bound with equality.

   Why: writing the unit candidate in A's own orthonormal eigenbasis makes \lVert r\rVert^2 a sum of squared eigenvalue gaps, and the smallest gap can only be smaller than that whole sum.

4. 4

   A small residual no longer guarantees anything for a non-normal A. Check instead whether A resembles a nilpotent Jordan block perturbed by a small entry \epsilon in one corner.

   • A is (or closely resembles) an n\times n nilpotent Jordan block with a small entry \epsilon in its lower-left corner — continue to Step 5.
   • A's non-normality has no such clean block structure — stop here; no simple formula applies, and the residual alone tells you nothing reliable.

   Why: the amplification this recipe warns about is specifically a property of nearly-defective structures like Jordan blocks, not of non-normal matrices in general.

5. 5

   For an n\times n nilpotent Jordan block perturbed by \epsilon in the lower-left corner, the perturbed eigenvalues satisfy \lambda^n=\epsilon, so |\lambda|=\epsilon^{1/n}.

   Example

   For n=3 and \epsilon=10^{-6}: \lambda=(10^{-6})^{1/3}=10^{-2}, so |\lambda|=0.01, ten thousand times larger than \epsilon itself.

   Why: forcing the characteristic polynomial of this specific perturbed block reduces directly to \lambda^n=\epsilon, and an n-th root of a tiny number is comparatively huge whenever n>1.