01 Key Concepts
Eigenvalue Definition
A scalar 'lambda' such that A*v = lambda*v for some nonzero vector v; the matrix scales v without changing its direction.
The Characteristic Equation
Eigenvalues are found by solving det(A - lambda*I) = 0, where I is the identity matrix of the same size as A.
Finding 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.
Trace 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).
Multiple Eigenvalues
An n x n matrix has exactly n eigenvalues (counting repeats), though some may be equal or complex.
02 Key Formulas
- det(A - lambda*I) = 0
- For 2x2: lambda^2 - (trace)*lambda + det(A) = 0
03 Solved Examples
Example 1 Find the eigenvalues of [[4,1],[2,3]].
- Trace = 4+3=7. Determinant = 4*3-1*2=12-2=10.
- Characteristic equation: lambda^2 - 7*lambda + 10 = 0.
- Factor: (lambda-2)(lambda-5)=0.
Answer: lambda = 2 or lambda = 5
Example 2 Find the eigenvalues of [[5,0],[0,3]] (a diagonal matrix).
- For a diagonal matrix, the eigenvalues are simply the diagonal entries themselves.
Answer: lambda = 5 and lambda = 3
Example 3 A 2x2 matrix has trace 6 and determinant 8. Find its eigenvalues.
- Characteristic equation: lambda^2 - 6*lambda + 8 = 0.
- Factor: (lambda-2)(lambda-4)=0.
Answer: lambda = 2 or lambda = 4
04 Practice Questions
1Find the eigenvalues of [[6,0],[0,2]].
6 and 2
2Find the trace and determinant of [[3,1],[1,3]].
Trace=6, Determinant=8
3Using the previous result, find the eigenvalues.
lambda^2-6lambda+8=0, so lambda=2 or 4
4How many eigenvalues does a 3x3 matrix have (counting repeats)?
3
5What equation is used to find eigenvalues?
det(A - lambda*I) = 0
📄 Eigenvalues — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.