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$(\begin{array}{cc} 2 & 1 \\ 0 & 2 \end{array})$ has algebraic multiplicity two but geometric multiplicity one at $2$

Example

For

$A = (\begin{array}{cc} 2 & 1 \\ 0 & 2 \end{array}) \in M_{2} (R),$

the eigenvalue $2$ has algebraic multiplicity $2$ and geometric multiplicity $1$ .

Facts & Assumptions

Given: The displayed real matrix $A$ .

[L1]

Algebraic multiplicity is the exponent of $x - λ$ in the characteristic polynomial, and geometric multiplicity is $\dim \ker (A - λ I)$ (Algebraic multiplicity as the exponent of $x - λ$ in $χ_{T}$ , and geometric multiplicity as $\dim E_{λ} (T)$ ).

Verification

technique · direct computation

1.1

One has $χ_{A} (x) = \det (\begin{array}{cc} x - 2 & - 1 \\ 0 & x - 2 \end{array}) = (x - 2)^{2}$ , so [L1] gives algebraic multiplicity $2$ .

L1algebra

1.2

Since $(A - 2 I) (u, v) = (v, 0)$ , its kernel is ${(u, 0) : u \in R}$ , a one-dimensional space.

L1algebra

2.1

Thus the geometric multiplicity is $1 < 2$ , so the general multiplicity inequality can be strict.

step 1.1step 1.2L1∎

Depends on

Algebraic multiplicity as the exponent of $x-\lambda$ in $\chi_T$, and geometric multiplicity as $\dim E_\lambda(T)$

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

H. Pinkham, Linear Algebra, §12.2 (standard reference, not scraped)

(2102)(20​12​) has algebraic multiplicity two but geometric multiplicity one at 22