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The real quarter-turn diagonalises after complexification but has no real eigenvector

Example

Let $T : R^{2} \to R^{2}$ be the quarter-turn with matrix

$A = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array})$

in the standard basis. Its complexification $T_{C}$ acts on $C^{2}$ by the same matrix, has eigenvalues $i$ and $- i$ with eigenvectors $(1, - i)$ and $(1, i)$ , and is therefore diagonalised over $C$ by that eigenbasis. Nevertheless $T$ itself has no real eigenvector.

Facts & Assumptions

Given: The quarter-turn $T$ with the displayed matrix $A$ .

[L1]

Complexification preserves the characteristic and minimal polynomials of a finite-dimensional real operator (Complexification preserves the characteristic and minimal polynomials of a finite-dimensional real operator).

[L2]

The canonical conjugation interchanges the generalised eigenspaces of $λ$ and $\overline{λ}$ (For a real operator, nonreal generalised eigenspaces of the complexification occur in conjugate pairs).

Verification

technique · direct

1.1

The characteristic polynomial is $χ_{T} (x) = \det (x I - A) = x^{2} + 1$ , and by [L1] the complexified operator has the same polynomial, which factors as $(x - i) (x + i)$ over $C$ .

L1algebra

1.2

For $λ = i$ , the equation $(A - i I) w = 0$ is $- i x - y = 0$ and $x - i y = 0$ , so $y = - i x$ and $w_{i} = (1, - i)$ is an eigenvector; symmetrically $w_{- i} = (1, i)$ is an eigenvector for $λ = - i$ .

algebra

2.1

The vectors $w_{i}$ and $w_{- i}$ are complex-linearly independent, so they form a complex basis of $C^{2}$ in which $T_{C}$ has the diagonal matrix $diag (i, - i)$ .

step 1.2algebra

2.2

Conjugation satisfies $σ (w_{i}) = (1, i) = w_{- i}$ , the conjugate-pair behaviour recorded in [L2] with exponent $1$ .

L2step 1.2

2.3

A real eigenvector $v \neq 0$ would carry a real eigenvalue $λ$ with $A v = λ v$ ; taking a nonzero coordinate of $v$ shows $λ$ is real, and then step 1.1 gives $λ^{2} + 1 = 0$ , which has no real solution.

step 1.1algebra

3.1

Steps 2.1 and 2.3 together prove the example: diagonalisation after complexification with no real eigenvector beforehand.

step 2.1step 2.3∎

Depends on

Used by

FALSE: complexification creates a real eigenvector whenever it creates a complex one False statement

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Sources

Keith Conrad, Complexification (notes) (standard reference, not scraped)