Linear Algebra · Quick Reference

Eigenvalues Cheat Sheet

Linear Algebra · Lesson 4/7

In one line: an eigenvalue tells you how much a special direction gets stretched or shrunk when a matrix transforms it — these special values unlock deep insight into a matrix's behavior.

Key Ideas

1Eigenvalue Definition. A scalar 'lambda' such that A*v = lambda*v for some nonzero vector v; the matrix scales v without changing its direction.

2The Characteristic Equation. Eigenvalues are found by solving det(A - lambda*I) = 0, where I is the identity matrix of the same size as A.

3Finding Eigenvalues for a 2x2 Matrix. For A=[[a,b],[c,d]], the characteristic equation becomes lambda^2 - (a+d)*lambda + (ad-bc) = 0, a quadratic in lambda.

4Trace and Determinant Shortcuts. The sum of eigenvalues always equals the trace of A (sum of diagonal entries); the product of eigenvalues always equals det(A).

5Multiple Eigenvalues. An n x n matrix has exactly n eigenvalues (counting repeats), though some may be equal or complex.

Worked Examples

Find the eigenvalues of [[4,1],[2,3]].

lambda = 2 or lambda = 5

Find the eigenvalues of [[5,0],[0,3]] (a diagonal matrix).

lambda = 5 and lambda = 3

A 2x2 matrix has trace 6 and determinant 8. Find its eigenvalues.

lambda = 2 or lambda = 4

Formulas

det(A - lambda*I) = 0

For 2x2: lambda^2 - (trace)*lambda + det(A) = 0

Practice Yourself

Find the eigenvalues of [[6,0],[0,2]].

6 and 2

Find the trace and determinant of [[3,1],[1,3]].

Trace=6, Determinant=8

Using the previous result, find the eigenvalues.

lambda^2-6lambda+8=0, so lambda=2 or 4