MAT018 · Matrices & determinants · Class 12 + B.Tech
Eigenvalues introduction
An eigenvalue λ of A is a number for which some non-zero vector v is only stretched, not turned: Av = λv. They solve det(A − λI) = 0.
λEigenvalues introduction
📐 Formula
det(A − λI) = 0; λ₁ + λ₂ = trace, λ₁λ₂ = det
✏️ Worked example
[2 1; 1 2]: λ² − 4λ + 3 = 0, so λ = 3 and 1. The circle becomes an ellipse stretched 3× and 1× along the eigen-directions.
🌍 Where it is used
Vibrations, Google's PageRank, stability of systems.
❓ Quick check
Eigenvalues of [5 0; 0 2]?
🏋️ Practice
EasySum of eigenvalues of [2 1; 1 2]?
Answer: 4 (the trace)
MediumProduct of eigenvalues of [2 1; 1 2]?
Answer: 3 (the determinant)
ChallengeEigenvalues of the identity?
Answer: 1 and 1