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Part CCSIR NET December 2023

Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

Let A be an n × n real symmetric matrix. Which of the following statements are necessarily true?

A.A is diagonalizable.✓
B.If Aᵏ = I for some positive integer k, then $A^{2} = I$ .✓
C.If Aᵏ = 0 for some positive integer k, then $A^{2} = 0$ .✓
D.All eigenvalues of A are real.✓

Solution

Spectral theorem: A = QDQᵀ with real D. Aᵏ = I forces each real eigenvalue to satisfy $λ^{k} = 1$ , so $λ = \pm 1$ and $A^{2} = I; A^{k} = 0$ forces all $λ = 0$ so A = 0.

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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