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MathidiaMathematics Academy
Numerical algorithms
Lesson 68 of 75
Standard practice

Eigenvalues and eigenvectors

Objective. Solve characteristic equations and find invariant directions.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

An eigenvector of a matrix M is a non-zero vector v satisfying Mv = λv for some scalar λ, the eigenvalue, meaning the matrix only scales v rather than changing its direction. Eigenvalues are found by solving det(M − λI) = 0, the characteristic equation, and corresponding eigenvectors are found by substituting each λ back into (M−λI)v=0 and solving. For example, for M=[[4,1],[2,3]], the characteristic equation (4−λ)(3−λ)−2=0 simplifies to λ²−7λ+10=0, giving eigenvalues λ=2 and λ=5.

Worked example

Find the eigenvalues of the matrix M = [[5,2],[2,2]].

5 … 2
Illustrated card showing 5 … 2
  1. 1Form the characteristic equation: det(M−λI) = (5−λ)(2−λ) − 4 = 0
  2. 2Expand: 10 − 5λ − 2λ + λ² − 4 = 0, giving λ² − 7λ + 6 = 0
  3. 3Factorise: (λ−1)(λ−6) = 0
  4. 4So the eigenvalues are λ=1 and λ=6

Answer: λ = 1 and λ = 6

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Common mistake. Students solve det(M−λI)=0 correctly but then substitute λ into M itself rather than M−λI when finding eigenvectors, producing nonsensical equations. Always form M−λI first, then solve (M−λI)v=0 for the eigenvector.

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