Eigenvalues and eigenvectors describe the special directions of a linear transformation. When a matrix transforms space, most vectors change both length and direction, but an eigenvector stays on its original line. It may stretch, shrink, or reverse direction, and the number that tells how much is the eigenvalue.
This idea matters in physics, engineering, computer graphics, data science, and differential equations because it reveals the natural directions of a system.
For a square matrix A, an eigenvector v and eigenvalue λ satisfy A v = λ v, where v is not the zero vector. To find eigenvalues, solve the characteristic equation det(A - λI) = 0, then substitute each λ into (A - λI)v = 0 to find its eigenvectors. Geometrically, eigenvectors are invariant directions under the transformation, while eigenvalues are scale factors along those directions.
In a 2 by 2 example, A = [[3, 1], [0, 2]] has characteristic equation (3 - λ)(2 - λ) = 0, so λ = 3 and λ = 2.
Key Facts
- Eigenvalue equation: A v = λ v, where v ≠ 0.
- Characteristic equation: det(A - λI) = 0.
- For a 2 by 2 matrix A = [[a, b], [c, d]], det(A - λI) = (a - λ)(d - λ) - bc.
- If λ > 1, the eigenvector is stretched; if 0 < λ < 1, it is shrunk; if λ < 0, it is reversed and scaled.
- For A = [[3, 1], [0, 2]], the eigenvalues are λ = 3 and λ = 2.
- Eigenvectors for different eigenvalues are linearly independent.
Vocabulary
- Eigenvalue
- An eigenvalue is the scalar λ that tells how much an eigenvector is scaled by a linear transformation.
- Eigenvector
- An eigenvector is a nonzero vector that stays on the same line after a matrix transformation.
- Characteristic equation
- The characteristic equation det(A - λI) = 0 is the equation used to find the eigenvalues of a square matrix.
- Linear transformation
- A linear transformation is a rule that maps vectors to vectors while preserving vector addition and scalar multiplication.
- Eigenspace
- An eigenspace is the set of all eigenvectors for one eigenvalue, together with the zero vector.
Common Mistakes to Avoid
- Using the zero vector as an eigenvector is wrong because A0 = λ0 is true for every λ and gives no special direction.
- Solving det(A) - λI = 0 is wrong because the determinant must be taken after subtracting λI, so the correct form is det(A - λI) = 0.
- Assuming every vector is an eigenvector is wrong because most transformed vectors change direction and do not remain on the same line.
- Forgetting to find eigenvectors after eigenvalues is incomplete because λ only gives the scale factor, while the eigenvector gives the actual invariant direction.
Practice Questions
- 1 Find the eigenvalues of A = [[4, 0], [0, 7]]. Then state one eigenvector for each eigenvalue.
- 2 For A = [[2, 1], [0, 3]], solve det(A - λI) = 0 and find an eigenvector for each eigenvalue.
- 3 A transformation sends one vector to twice itself and sends another vector to a vector pointing in a different direction. Explain which vector is an eigenvector and why.