Characteristic Polynomial Definition and Properties | Generated by AI
In linear algebra, the characteristic polynomial of a square matrix \( A \) (of size \( n \times n \)) is a polynomial that encodes information about the matrix’s eigenvalues. It is defined as:
\[ p(\lambda) = \det(A - \lambda I) \]
where:
- \( A \) is the \( n \times n \) matrix.
- \( \lambda \) is a scalar (the variable of the polynomial).
- \( I \) is the \( n \times n \) identity matrix.
- \( \det \) denotes the determinant.
Key Characteristics
- Form: The characteristic polynomial is a monic polynomial of degree \( n \), meaning its leading coefficient (for the \( \lambda^n \) term) is 1.
- For a \( 2 \times 2 \) matrix \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the characteristic polynomial is: \[ p(\lambda) = \det \begin{bmatrix} a - \lambda & b \ c & d - \lambda \end{bmatrix} = \lambda^2 - (a + d)\lambda + (ad - bc) \] where \( a + d \) is the trace and \( ad - bc \) is the determinant.
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Eigenvalues: The roots of the characteristic polynomial \( p(\lambda) = 0 \) are the eigenvalues of the matrix \( A \). These can be real or complex numbers.
- Coefficients: The coefficients of the polynomial are related to the matrix’s properties:
- The coefficient of \( \lambda^{n-1} \) is \( -\text{trace}(A) \).
- The constant term is \( (-1)^n \det(A) \).
- Other coefficients are determined by sums of principal minors of \( A \).
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Invariance: The characteristic polynomial is invariant under similarity transformations. If \( B = P^{-1}AP \), then \( A \) and \( B \) have the same characteristic polynomial.
- Cayley-Hamilton Theorem: The matrix \( A \) satisfies its own characteristic polynomial, i.e., if \( p(\lambda) \) is the characteristic polynomial, then \( p(A) = 0 \) (the zero matrix).
Example
For a matrix \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \):
- Compute \( A - \lambda I = \begin{bmatrix} 1 - \lambda & 2 \ 3 & 4 - \lambda \end{bmatrix} \).
- The characteristic polynomial is: \[ p(\lambda) = \det \begin{bmatrix} 1 - \lambda & 2 \ 3 & 4 - \lambda \end{bmatrix} = (1 - \lambda)(4 - \lambda) - (2)(3) = \lambda^2 - 5\lambda + 4 - 6 = \lambda^2 - 5\lambda - 2 \]
- The eigenvalues are found by solving \( \lambda^2 - 5\lambda - 2 = 0 \).
Applications
- Eigenvalue computation: Solving \( p(\lambda) = 0 \) gives the eigenvalues.
- Matrix properties: Helps determine diagonalizability, invertibility, and other properties.
- Differential equations: Used in systems of linear differential equations to find solutions.
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