Linear Algebra · Lesson 10
Eigenvalues & Eigenvectors
An eigenvector is a special direction that a matrix transformation doesn't rotate — only scales. The scale factor is the eigenvalue. Eigenanalysis is arguably the most important concept in applied linear algebra, powering PCA, Google PageRank, and quantum mechanics.
Quick Check
1. If Av = 3v, then v is an eigenvector with eigenvalue:
v
3
1/3
A
2. The characteristic equation for eigenvalues is:
Av = λv directly
det(A − λI) = 0
trace(A) = 0
det(A) = λ
3. For A = [[2,0],[0,5]], the eigenvalues are:
2 and −5
2 and 5
7 and 10
√2 and √5
4. For a 2×2 matrix, the product of eigenvalues equals:
The trace
The determinant
The sum of all entries
0