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Part BCSIR NET June 2023algebraic-vs-geometric-multiplicity

Algebraic vs geometric multiplicity

Let A be a 3×3 real matrix whose characteristic polynomial p(T) is divisible by $T^{2}$ . Which of the following statements is true?

A.The eigenspace of A for the eigenvalue 0 is two-dimensional.
B.All the eigenvalues of A are real.✓
C. $A^{3} = 0$ .
D.A is diagonalizable.

Solution

$p (T) = T^{2} (T - c)$ with c real (degree-3 real polynomial). So all eigenvalues are real. The geometric multiplicity of 0 may be 1 (Jordan block), so (1), (4) fail; A need not be nilpotent (c ≠ 0).

The trap it tests

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this

Related counterexample: Same characteristic polynomial ⇒ similar

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From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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