Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-13
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A quarter-turn of R2 has characteristic polynomial x2+1 and no real eigenvalue

Example

The real quarter-turn

A=(0−110)

has χA(x)=x2+1 and empty real spectrum.

Facts & Assumptions

Given: The displayed matrix, acting on R2.

[L1]

The spectrum over the base field is exactly the root set in that field of the characteristic polynomial (For every finite-dimensional space, σF(T) is exactly the set of roots in F of χT).

[L2]

Every real square is nonnegative (Squares of nonzero elements are positive).

Verification

technique · direct computation
1.1

Directly, χA(x)=det⁡(x1−1x)=x2+1.

algebra
2.1

For every x∈R, [L2] gives x2+1≥1>0, so χA has no real root.

step 1.1L2algebra
3.1

By [L1], A has no real eigenvalue, despite acting on a nonzero real space.

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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