Overview
The characteristic polynomial is a polynomial whose roots are the eigenvalues of a matrix. It encodes essential information about the matrix's spectral properties.
Definition
For an n × n n \times n n × n A A A
p ( λ ) = det ( A − λ I ) p(\lambda) = \det(A - \lambda I) p ( λ ) = det ( A − λ I )
This is a polynomial of degree n n n λ \lambda λ
Standard Form
For an n × n n \times n n × n
p ( λ ) = ( − 1 ) n λ n + ( − 1 ) n − 1 ( trace A ) λ n − 1 + ⋯ + det ( A ) p(\lambda) = (-1)^n\lambda^n + (-1)^{n-1}(\text{trace } A)\lambda^{n-1} + \cdots + \det(A) p ( λ ) = ( − 1 ) n λ n + ( − 1 ) n − 1 ( trace A ) λ n − 1 + ⋯ + det ( A )
Or equivalently:
p ( λ ) = ( λ 1 − λ ) ( λ 2 − λ ) ⋯ ( λ n − λ ) p(\lambda) = (\lambda_1 - \lambda)(\lambda_2 - \lambda)\cdots(\lambda_n - \lambda) p ( λ ) = ( λ 1 − λ ) ( λ 2 − λ ) ⋯ ( λ n − λ )
where λ 1 , λ 2 , … , λ n \lambda_1, \lambda_2, \ldots, \lambda_n λ 1 , λ 2 , … , λ n
Computing for 2×2 Matrices
A = [ a b c d ] A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} A = [ a c b d ]
p ( λ ) = det [ a − λ b c d − λ ] = ( a − λ ) ( d − λ ) − b c p(\lambda) = \det\begin{bmatrix} a-\lambda & b \\ c & d-\lambda \end{bmatrix} = (a-\lambda)(d-\lambda) - bc p ( λ ) = det [ a − λ c b d − λ ] = ( a − λ ) ( d − λ ) − b c
= λ 2 − ( a + d ) λ + ( a d − b c ) = λ 2 − ( trace ) λ + det = \lambda^2 - (a+d)\lambda + (ad-bc) = \lambda^2 - (\text{trace})\lambda + \det = λ 2 − ( a + d ) λ + ( a d − b c ) = λ 2 − ( trace ) λ + det
Computing for 3×3 Matrices
A = [ a b c d e f g h i ] A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} A = a d g b e h c f i
p ( λ ) = − λ 3 + ( trace ) λ 2 − ( sum of 2 × 2 principal minors ) λ + det ( A ) p(\lambda) = -\lambda^3 + (\text{trace})\lambda^2 - (\text{sum of } 2 \times 2 \text{ principal minors})\lambda + \det(A) p ( λ ) = − λ 3 + ( trace ) λ 2 − ( sum of 2 × 2 principal minors ) λ + det ( A )
Example: 2×2 Matrix
A = [ 4 2 1 3 ] A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix} A = [ 4 1 2 3 ]
p ( λ ) = λ 2 − ( 4 + 3 ) λ + ( 4 × 3 − 2 × 1 ) = λ 2 − 7 λ + 10 = ( λ − 5 ) ( λ − 2 ) p(\lambda) = \lambda^2 - (4+3)\lambda + (4 \times 3 - 2 \times 1) = \lambda^2 - 7\lambda + 10 = (\lambda-5)(\lambda-2) p ( λ ) = λ 2 − ( 4 + 3 ) λ + ( 4 × 3 − 2 × 1 ) = λ 2 − 7 λ + 10 = ( λ − 5 ) ( λ − 2 )
Eigenvalues: λ = 5 \lambda = 5 λ = 5 λ = 2 \lambda = 2 λ = 2
Example: 3×3 Matrix
A = [ 2 0 0 0 3 1 0 1 3 ] A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 3 \end{bmatrix} A = 2 0 0 0 3 1 0 1 3
A − λ I = [ 2 − λ 0 0 0 3 − λ 1 0 1 3 − λ ] A - \lambda I = \begin{bmatrix} 2-\lambda & 0 & 0 \\ 0 & 3-\lambda & 1 \\ 0 & 1 & 3-\lambda \end{bmatrix} A − λ I = 2 − λ 0 0 0 3 − λ 1 0 1 3 − λ
p ( λ ) = ( 2 − λ ) [ ( 3 − λ ) 2 − 1 ] = ( 2 − λ ) ( λ 2 − 6 λ + 8 ) = ( 2 − λ ) ( λ − 4 ) ( λ − 2 ) p(\lambda) = (2-\lambda)[(3-\lambda)^2 - 1] = (2-\lambda)(\lambda^2 - 6\lambda + 8) = (2-\lambda)(\lambda-4)(\lambda-2) p ( λ ) = ( 2 − λ ) [( 3 − λ ) 2 − 1 ] = ( 2 − λ ) ( λ 2 − 6 λ + 8 ) = ( 2 − λ ) ( λ − 4 ) ( λ − 2 )
Eigenvalues: λ = 2 \lambda = 2 λ = 2 λ = 4 \lambda = 4 λ = 4
Cayley-Hamilton Theorem
Every matrix satisfies its own characteristic polynomial:
p ( A ) = 0 p(A) = 0 p ( A ) = 0
If p ( λ ) = λ 2 − 7 λ + 10 p(\lambda) = \lambda^2 - 7\lambda + 10 p ( λ ) = λ 2 − 7 λ + 10
A 2 − 7 A + 10 I = O A^2 - 7A + 10I = O A 2 − 7 A + 10 I = O
Applications
Computing matrix powers: A 2 = 7 A − 10 I A^2 = 7A - 10I A 2 = 7 A − 10 I
Finding matrix inverse: A − 1 = ( 7 I − A ) / 10 A^{-1} = (7I - A)/10 A − 1 = ( 7 I − A ) /10
Properties from Characteristic Polynomial
From p ( λ ) = λ n − c 1 λ n − 1 + c 2 λ n − 2 − ⋯ + ( − 1 ) n c n p(\lambda) = \lambda^n - c_1\lambda^{n-1} + c_2\lambda^{n-2} - \cdots + (-1)^n c_n p ( λ ) = λ n − c 1 λ n − 1 + c 2 λ n − 2 − ⋯ + ( − 1 ) n c n
Property Value Sum of eigenvalues c 1 = trace ( A ) c_1 = \text{trace}(A) c 1 = trace ( A ) Product of eigenvalues c n = det ( A ) c_n = \det(A) c n = det ( A )
Factoring p ( λ ) p(\lambda) p ( λ )
Strategies
For triangular matrices : eigenvalues are diagonal entriesFactor by grouping : for special structuresRational root theorem : test integer factors of constant termQuadratic formula : for 2 × 2 2 \times 2 2 × 2
Complex Eigenvalues
For real matrices, complex eigenvalues come in conjugate pairs.
A = [ 0 1 − 1 0 ] A = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} A = [ 0 − 1 1 0 ]
p ( λ ) = λ 2 + 1 = ( λ − i ) ( λ + i ) p(\lambda) = \lambda^2 + 1 = (\lambda - i)(\lambda + i) p ( λ ) = λ 2 + 1 = ( λ − i ) ( λ + i )
Eigenvalues: λ = ± i \lambda = \pm i λ = ± i
Eigenvalue Bounds
The eigenvalues satisfy:
∣ λ ∣ ≤ ∥ A ∥ \lvert\lambda\rvert \leq \|A\| ∣ λ ∣ ≤ ∥ A ∥ Gershgorin circle theorem gives tighter bounds