Characteristic Polynomial Calculator

What is a Characteristic Polynomial?

The characteristic polynomial of a square matrix $A$ is a polynomial whose roots are the matrix's eigenvalues ($\lambda$). In linear algebra, eigenvalues and eigenvectors provide deep insights into matrix transformations, stability of dynamical systems, quantum mechanics, and structural vibrations.

Definition of Characteristic Polynomial:

$$p(\lambda) = \det(\lambda I - A) \quad \text{or} \quad p(\lambda) = \det(A - \lambda I)$$

Where $I$ is the identity matrix of the same dimension as $A$, and $\det$ represents the matrix determinant.

Characteristic Polynomial Formulas by Matrix Size

1. For a $2 \times 2$ Matrix

Consider a general $2 \times 2$ matrix:

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

The characteristic polynomial can be directly derived without expanding long determinants:

$$p(\lambda) = \lambda^2 - \text{tr}(A)\lambda + \det(A)$$

Trace: $\text{tr}(A) = a + d$ (sum of main diagonal elements)
Determinant: $\det(A) = ad - bc$

2. For a $3 \times 3$ Matrix

For a $3 \times 3$ matrix:

$$A = \begin{pmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{pmatrix}$$

The monic characteristic polynomial is given by:

$$p(\lambda) = \lambda^3 - \text{tr}(A)\lambda^2 + M\lambda - \det(A)$$

Trace: $\text{tr}(A) = a_{11} + a_{22} + a_{33}$
Sum of Principal $2 \times 2$ Minors ($M$): $$M = (a_{22}a_{33} - a_{23}a_{32}) + (a_{11}a_{33} - a_{13}a_{31}) + (a_{11}a_{22} - a_{12}a_{21})$$
Determinant: $\det(A)$ computed by Laplace expansion

Eigenvalues and the Characteristic Equation

Setting the characteristic polynomial to zero gives the characteristic equation:

$$p(\lambda) = 0$$

The roots of this equation are the matrix's eigenvalues. Depending on the discriminant, these roots may be:

Distinct Real Roots: Corresponding to real scaling axes.
Repeated Roots: Indicating algebraic multiplicity greater than 1.
Complex Conjugate Pairs ($u \pm vi$): Indicating rotational components in the linear transformation.

The Cayley-Hamilton Theorem

An important theorem in matrix theory is the Cayley-Hamilton theorem, which states that every square matrix satisfies its own characteristic polynomial:

$$p(A) = 0$$

This allows high powers of matrices $A^k$ and matrix inverses $A^{-1}$ to be calculated easily as lower-degree polynomial expressions in $A$.

Related Matrix & Algebra Tools

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Frequently Asked Questions

What is the difference between $\det(A - \lambda I)$ and $\det(\lambda I - A)$?

Both definitions are widely used in linear algebra. They differ at most by a sign factor of $(-1)^n$, where $n$ is the dimension of the matrix. For an $n \times n$ matrix, both yield the exact same roots (eigenvalues).

Can a real matrix have complex eigenvalues?

Yes. If the characteristic polynomial has a negative discriminant, the roots will occur as complex conjugate pairs ($a \pm bi$), which corresponds geometrically to rotation and scaling in the vector space.

How is the trace of a matrix related to its eigenvalues?

The sum of all eigenvalues of a matrix equals the trace of the matrix: $\sum \lambda_i = \text{tr}(A)$. Similarly, the product of all eigenvalues equals the determinant: $\prod \lambda_i = \det(A)$.

Do similar matrices have the same characteristic polynomial?

Yes. If two matrices $A$ and $B$ are similar (meaning $B = P^{-1}AP$ for an invertible matrix $P$), they share the identical characteristic polynomial, trace, determinant, and eigenvalues.

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