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Concept guide



## [Eigenvalues of 2×2 real matrices: characteristic polynomial, discriminant and nonzero eigenvectors](/en/derivations/derivation-mko-eigenvalue-01/?from=%2Fen%2Fderivations%2F%3Fq%3Dmko-eigenvalue)

For 2×2 real matrices prove: λ is a real eigenvalue ⇔ det(M−λI)=0 ⇔ λ²−tr(M)λ+det(M)=0, and the number of distinct real eigenvalues is decided by the sign of Δ=tr(M)²−4det(M) (negative, zero, positive give 0, 1, 2). For M=[[4,1],[2,3]] find all distinct real eigenvalues with nonzero eigenvectors, show that for the candidate λ=3 the kernel is K₃={0} and 3 is not an eigenvalue, and reject the candidate eigenpair (3,0); then contrast with the rotation R=[[0,−1],[1,0]] (Δ<0) and J=[[3,1],[0,3]] (Δ=0).

Real eigenvalues and nonzero eigenvectors of 2×2 matrices

derivation-mko-eigenvalue-01





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