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The scalar matrix $2 I_{2}$ has algebraic and geometric multiplicity two at $2$

Example

Over any field $F$ , the scalar matrix $A = 2 I_{2}$ has algebraic and geometric multiplicity $2$ at the scalar $2 \in F$ .

Facts & Assumptions

Given: The matrix $A = 2 I_{2} \in M_{2} (F)$ .

[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 $\dim E_{λ} (T)$ ).

[L2]

The two standard coordinate vectors form a basis of $F^{2}$ (The standard list $e : n \to F^{n}$ with $e_{i} (i) = 1_{F}$ and $e_{i} (j) = 0_{F}$ for $j \neq i$ is an ordered basis of $F^{n}$ ; hence $\dim_{F} F^{n} = n$ , and $F^{0}$ is the zero space with basis $\emptyset$ and dimension $0$ ).

Verification

technique · direct computation

1.1

Since $x I_{2} - A = (x - 2) I_{2}$ , one has $χ_{A} (x) = (x - 2)^{2}$ , so the algebraic multiplicity is $2$ .

L1algebra

1.2

Since $A - 2 I_{2} = 0$ , its kernel is all of $F^{2}$ , which has dimension $2$ by [L2]. Thus the geometric multiplicity is $2$ .

L1L2algebra

2.1

Both multiplicities therefore equal $2$ , including in characteristic $2$ , where the scalar denoted $2$ is $0$ .

step 1.1step 1.2∎

Depends on

Used by

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Sources

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

The scalar matrix 2I22I2​ has algebraic and geometric multiplicity two at 22