ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

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Different gauge choices change the eigenvector derivative but not the eigenvalue derivative or projector derivative

Example

Let

$A (t) = (\begin{array}{cc} 1 & t \\ 0 & 2 \end{array}) .$

For the eigenvalue $2$ , one right eigenvector branch is $x_{1} (t) = (t, 1)^{T}$ , which fixes the second component at $1$ ; another is

$x_{2} (t) = \frac{1}{1 + t} (t, 1)^{T},$

which fixes the sum of the components at $1$ . Then

$x_{1}^{'} (0) = (\begin{array}{c} 1 \\ 0 \end{array}), x_{2}^{'} (0) = (\begin{array}{c} 1 \\ - 1 \end{array}) = x_{1}^{'} (0) - x_{1} (0) .$

So the eigenvector derivative depends on the gauge, even though the eigendirection and the spectral projector do not.

Facts & Assumptions

Given: The family $A (t)$ and the eigenvalue branch $λ (t) \equiv 2$ .

[L1]

Different fixed gauges give eigenvector derivatives determined only up to addition of a multiple of the eigenvector, while the spectral projector derivative is gauge-invariant (In a fixed gauge, the derivative of a simple right eigenvector is obtained by applying the reduced resolvent to the perturbation, The derivative of the simple spectral projector is expressed by the reduced resolvent and the perturbation).

Verification

technique · direct

1.1

Direct multiplication shows $A (t) (t, 1)^{T} = 2 (t, 1)^{T}$ , so both displayed branches are valid right eigenvectors for $λ = 2$ . Differentiating gives the displayed derivatives at $t = 0$ .

algebra

2.1

The difference is $x_{2}^{'} (0) - x_{1}^{'} (0) = - x_{1} (0)$ , a multiple of the eigenvector itself. Thus the eigenvector derivative depends on the normalization, exactly as [L1] warns. Both branches span the same eigendirection, so they define the same spectral projector.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

G. W. Stewart and Ji-guang Sun, Matrix Perturbation Theory (standard reference, not scraped)