Lesson 2: The Characteristic Equation
Estimated time: 40-50 minutes
Learning Objectives
- Derive the characteristic polynomial from det(A - lambda I) = 0
- Compute the characteristic polynomial for 2x2 and 3x3 matrices
- Define and distinguish algebraic and geometric multiplicity
- Use the characteristic equation to find all eigenvalues
The Characteristic Polynomial
The characteristic equation det(A - lambda I) = 0 produces a polynomial in lambda. For an n x n matrix, this is a degree-n polynomial.
Characteristic Polynomial: p(lambda) = det(A - lambda I). This is a polynomial of degree n with leading term (-lambda)^n (or equivalently (-1)^n * lambda^n).
The 2x2 Case
2x2 Formula: For A = [a b; c d], the characteristic polynomial is:
p(lambda) = lambda^2 - (a+d)*lambda + (ad - bc) = lambda^2 - trace(A)*lambda + det(A).
Worked Example 1
A = [6 -1; 2 3]. trace = 9, det = 18 + 2 = 20.
p(lambda) = lambda^2 - 9*lambda + 20 = (lambda - 4)(lambda - 5) = 0.
Eigenvalues: lambda = 4 and lambda = 5.
Worked Example 2: Repeated Eigenvalue
A = [3 1; 0 3]. trace = 6, det = 9.
p(lambda) = lambda^2 - 6*lambda + 9 = (lambda - 3)^2 = 0.
Only eigenvalue: lambda = 3 with algebraic multiplicity 2.
The 3x3 Case
For 3x3 matrices, expand det(A - lambda I) using cofactor expansion. The result is a cubic polynomial.
Worked Example 3: 3x3 Characteristic Polynomial
A = [2 0 0; 0 3 -1; 0 -1 3].
A - lambda I = [2-lambda, 0, 0; 0, 3-lambda, -1; 0, -1, 3-lambda].
Expand along row 1: det = (2-lambda) * det([3-lambda, -1; -1, 3-lambda]).
= (2-lambda)[(3-lambda)^2 - 1] = (2-lambda)(lambda^2 - 6*lambda + 8) = (2-lambda)(lambda-2)(lambda-4).
Eigenvalues: lambda = 2 (algebraic multiplicity 2) and lambda = 4 (algebraic multiplicity 1).
Worked Example 4
A = [1 2 0; 0 3 0; 0 0 5].
Since A is upper triangular, det(A - lambda I) = (1-lambda)(3-lambda)(5-lambda) = 0.
Eigenvalues: lambda = 1, 3, 5 -- just read them from the diagonal!
Algebraic vs. Geometric Multiplicity
Algebraic Multiplicity: The number of times lambda appears as a root of the characteristic polynomial.
Geometric Multiplicity: The dimension of the eigenspace E_lambda = null(A - lambda I). This equals the number of free variables when you row reduce (A - lambda I).
Key Inequality
For every eigenvalue lambda: 1 ≤ geometric multiplicity ≤ algebraic multiplicity.
Worked Example 5: When Multiplicities Differ
A = [3 1; 0 3]. Eigenvalue lambda = 3 has algebraic multiplicity 2.
A - 3I = [0 1; 0 0]. Row reduce: [0 1; 0 0]. One free variable (x1). Geometric multiplicity = 1.
Eigenspace: E_3 = span{(1, 0)}. Only one linearly independent eigenvector despite algebraic multiplicity 2.
Worked Example 6: When Multiplicities Match
A = [3 0; 0 3] = 3I. Eigenvalue lambda = 3 has algebraic multiplicity 2.
A - 3I = [0 0; 0 0]. Two free variables. Geometric multiplicity = 2.
Eigenspace: E_3 = R^2 (every nonzero vector is an eigenvector). Multiplicities match!
Complex Eigenvalues
Not every real matrix has real eigenvalues. The characteristic polynomial may have complex roots.
Worked Example 7
A = [0 -1; 1 0] (90-degree rotation).
p(lambda) = lambda^2 + 1 = 0. Eigenvalues: lambda = i and lambda = -i.
No real eigenvectors exist -- a rotation does not preserve any real direction!
Complex Eigenvalues of Real Matrices: Complex eigenvalues always come in conjugate pairs: if lambda = a + bi is an eigenvalue, so is lambda-bar = a - bi.
Check Your Understanding
1. Find the characteristic polynomial of A = [2 5; 1 4].
2. A 3x3 matrix has characteristic polynomial (lambda - 1)^2(lambda + 3). What are the eigenvalues and their algebraic multiplicities?
3. For A = [3 1; 0 3], find the geometric multiplicity of lambda = 3.
Key Takeaways
- Characteristic polynomial: p(lambda) = det(A - lambda I) is degree n for an n x n matrix
- 2x2 shortcut: p(lambda) = lambda^2 - trace(A)*lambda + det(A)
- Triangular matrices: eigenvalues are diagonal entries
- Algebraic multiplicity = multiplicity as root of p(lambda); geometric multiplicity = dim(eigenspace)
- 1 ≤ geometric mult. ≤ algebraic mult. always
- Complex eigenvalues of real matrices come in conjugate pairs