Overview
This lecture introduces the fundamental concepts of eigenvalues and eigenvectors, explains their definitions and significance, and demonstrates methods for finding them using examples.
Definitions and Core Equation
- An eigenvector of a matrix A is a vector x such that A x = λ x for some scalar λ.
- The scalar λ is called the eigenvalue corresponding to the eigenvector x.
- For eigenvalue λ = 0, the eigenvectors are in the null space of A.
- Most vectors are not eigenvectors; eigenvectors are special as A x is parallel to x (possibly reversed or zero).
Examples and Applications
- Projection Matrix: Eigenvectors in the projection plane have eigenvalue 1; perpendicular vectors have eigenvalue 0.
- Permutation Matrix (e.g., [[0,1],[1,0]]): Eigenvalues are 1 and -1; corresponding eigenvectors are [1,1] and [-1,1].
- Trace: The sum of eigenvalues equals the trace (sum down the diagonal) of the matrix.
- Determinant: The product of eigenvalues equals the determinant of the matrix.
Finding Eigenvalues and Eigenvectors
- To find eigenvalues, solve det(A - λI) = 0 (the characteristic equation).
- The roots λ of the characteristic equation are eigenvalues.
- After finding λ, substitute into (A - λI)x = 0 to find the corresponding eigenvector(s).
- For 2x2 matrices, the characteristic equation is quadratic: λ² - (trace)λ + (determinant) = 0.
Special Cases and Properties
- Adding a scalar multiple of identity (kI) to A increases all eigenvalues by k; eigenvectors remain unchanged.
- In general, eigenvalues of A+B or A·B are not simple combinations of those of A and B, except for scalar multiples of identity.
- A rotation matrix (e.g., 90-degree) may have complex eigenvalues (e.g., i and -i) and no real eigenvectors.
- For triangular matrices, eigenvalues are the entries on the diagonal.
- Repeated eigenvalues can lead to a shortage of independent eigenvectors (defective matrices).
Key Terms & Definitions
- Eigenvector — A nonzero vector x where A x = λ x for some scalar λ.
- Eigenvalue (λ) — The scalar in the equation A x = λ x that scales the eigenvector.
- Null Space — Set of vectors x where A x = 0.
- Trace — The sum of diagonal entries of a matrix.
- Determinant — A scalar value representing the scaling factor of a matrix; equals the product of eigenvalues.
- Characteristic Equation — The equation det(A - λI) = 0 used to find eigenvalues.
- Defective Matrix — A matrix with fewer independent eigenvectors than its size.
Action Items / Next Steps
- Practice finding eigenvalues and eigenvectors for 2x2 and 3x3 matrices.
- Review properties of special matrices (projection, permutation, rotation, triangular).
- Prepare for further exploration of uses and deeper theory in the next lecture.