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Characteristic Polynomial Calculator
Calculate, understand, simplify, and apply characteristic polynomials for eigenvalues, matrix analysis, and linear algebra coursework. Use the tool below, then follow the guides for formulas, examples, and common mistakes.
- 2×2, 3×3, and 4×4 sizes
- det(A − λI) or det(λI − A)
- Runs privately in your browser
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Characteristic Polynomial Calculator
Enter integers, decimals, or fractions (example: 3/4). Fill every cell to update p(λ). Matrix entries use row letters with subscripts (a₁, b₂, …).
Matrix (2×2)
Characteristic polynomial
Using: det(A − λI)
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Fill all matrix entries to compute the characteristic polynomial.
How to use this calculator
- Select 2×2, 3×3, or 4×4.
- Choose det(A − λI) or det(λI − A).
- Type each value in the labeled grid (a₁, a₂, b₁, b₂, …).
- Read p(λ) in the result panel, or clear entries to start over.
Characteristic Polynomial Calculator
Enter integers, decimals, or fractions (example: 3/4). Fill every cell to update p(λ). Matrix entries use row letters with subscripts (a₁, b₂, …).
Matrix ()
Characteristic polynomial
Using:
Fill all matrix entries to compute the characteristic polynomial.
Use valid numbers or fractions (example: -2, 0.5, 3/4).
Roots of p(λ) are eigenvalues of A (over the complex numbers). For det(λI − A) and det(A − λI), the roots match; the polynomial may differ by an overall sign when n is odd.
How to use this calculator
- Select 2×2, 3×3, or 4×4.
- Choose det(A − λI) or det(λI − A).
- Type each value in the labeled grid (a₁, a₂, b₁, b₂, …).
- Read p(λ) in the result panel, or clear entries to start over.
What Is a Characteristic Polynomial?
The characteristic polynomial p(λ) of an n×n matrix A is the matrix polynomial defined by p(λ) = det(A − λI), where I is the identity matrix of the same size. You treat λ as an indeterminate, subtract it from each diagonal entry of A, and take the determinant of the resulting matrix.
In meaning, p(λ) packages all eigenvalue information: a scalar λ is an eigenvalue if and only if it is a root of p(λ). That is why the equation p(λ) = 0 is called the characteristic equation.
The object is central to linear algebra, differential equations, and engineering mathematics. In stability analysis, roots of p(λ) describe whether a dynamic system settles or grows. In diagonalization, factoring p(λ) guides whether A is similar to a simple diagonal matrix.
Use the calculator at the top of this page to expand p(λ) for 2×2, 3×3, or 4×4 matrices, then read what is a characteristic polynomial for a longer introduction.
Definition
p(λ) = det(A − λI). Some courses write det(λI − A); roots match, but the polynomial may differ by a sign factor.
Matrix polynomial idea
Entries of A − λI are linear in λ, so det(A − λI) is a polynomial of degree n in λ.
Eigenvalue link
Eigenvalues are roots of p(λ). Algebraic multiplicity counts how many times a root appears in the factorization.
Applications
Homework checks, exam preparation, control and vibration models, and verifying software linear algebra output.
Characteristic Polynomial Formula
p(λ) = det(A − λI)
p(λ) = det(λI − A) (same roots; may differ by (−1)^n overall sign)
A − λI: subtract λ from each diagonal entry of A
2×2 for A = [[a, b], [c, d]]:
p(λ) = λ² − (a + d)λ + (ad − bc)
Coefficient facts (det(A − λI) form):
deg p(λ) = n, constant term p(0) = det(A), coeff of λ^(n−1) involves −tr(A)
The determinant formula is the definition used in most introductory texts. Identity matrix I has ones on the diagonal, so A − λI is easy to write before expansion.
Polynomial expansion means multiplying factors from cofactor expansion or row reduction rules that preserve determinants, then collecting λ powers.
Matrix notation: if A = (aij), then (A − λI)ij = aij for i ≠ j and aii − λ on the diagonal. See the characteristic polynomial formula article for more identities.
How to Find the Characteristic Polynomial
Follow this step-by-step method by hand, then confirm with the calculator above. For a full walkthrough, read <a href="/blog/how-to-find-characteristic-polynomial/" class="text-link underline">how to find the characteristic polynomial</a>.
Confirm A is square (n×n)
Characteristic polynomials are defined only for square matrices. Label rows and columns consistently.
Construct A − λI
Subtract λ from each diagonal entry. Keep off-diagonal entries unchanged when using det(A − λI).
Compute det(A − λI)
Use cofactor expansion, row operations, or size shortcuts. Treat λ as a symbol, not a number.
Simplify the polynomial
Collect like terms into standard form. Check degree n and constant term det(A).
Extract eigenvalues (optional)
Solve p(λ) = 0 by factoring or numerical methods. Roots are eigenvalues over ℂ.
Characteristic Polynomial Examples
Work through these 2×2 and 3×3 examples in the calculator above, then read <a href="/blog/characteristic-polynomial-examples/" class="text-link underline">characteristic polynomial examples</a> for more setups.
2×2 symmetric matrix
A = [[2, 1], [1, 2]].
det(A − λI) = (2−λ)² − 1 = λ² − 4λ + 3
Polynomial: p(λ) = λ² − 4λ + 3; eigenvalues λ = 1, 3
3×3 diagonal matrix
A = diag(1, 2, 3).
Product (1−λ)(2−λ)(3−λ).
Polynomial: p(λ) = −λ³ + 6λ² − 11λ + 6
2×2 rotation-style
A = [[0, 1], [−1, 0]].
p(λ) = λ² + 1
Polynomial: Eigenvalues λ = ±i (complex roots)
Engineering stiffness sketch
A = [[4, 1], [2, 3]] models a small coupled system.
Expand det(A − λI) by hand or with the tool.
Polynomial: Use roots of p(λ) to discuss stability signs in a follow-up ODE model
Characteristic Polynomial and Eigenvalues
An eigenvalue λ satisfies Av = λv for some nonzero vector v. Equivalently, (A − λI)v = 0 has a nontrivial solution, so det(A − λI) = 0. Therefore eigenvalues are exactly the roots of the characteristic polynomial.
Algebraic multiplicity is the exponent of (λ − λ0) in the factored form of p(λ). It can exceed geometric multiplicity when defective matrices appear in advanced courses.
Matrix diagonalization is possible when A has enough linearly independent eigenvectors, which is tied to how p(λ) factors. Spectral interpretation: diagonal entries of a similar diagonal matrix are eigenvalues.
Read characteristic polynomial and eigenvalues for worked links between p(λ), eigenvectors, and diagonalization.
Characteristic equation
p(λ) = 0 is the equation whose solutions are eigenvalues.
Trace and determinant
For 2×2, sum of eigenvalues equals tr(A) and product equals det(A).
Complex roots
Even real matrices can have complex eigenvalues; p(λ) may be irreducible over ℝ.
Characteristic Equation Calculator
The characteristic equation is p(λ) = 0, where p(λ) = det(A − λI). This page calculator returns the expanded polynomial; solving p(λ) = 0 is the eigenvalue step.
For 2×2, the quadratic formula applies once p(λ) is written as aλ² + bλ + c. For 3×3, factoring or cubic methods may be required. For 4×4, numerical root finders are common in practice.
Algebraic simplification before solving saves effort: factor common terms and check rational roots when coefficients are integers.
See characteristic equation calculator for setup tips and links back to the matrix tool.
Characteristic equation: det(A − λI) = 0
Equivalent: p(λ) = 0 where p(λ) = det(A − λI)
Setup
Build p(λ) first with the calculator, then solve for λ.
Matrix roots
Roots are eigenvalues; they need not be real.
Numerical methods
When factoring is hard, use approved course tools for approximate roots.
Determinants and Characteristic Polynomials
The characteristic polynomial is defined through a determinant, so determinant rules (row swaps, scaling, adding multiples of rows) apply while you expand det(A − λI).
Polynomial degree is always n for an n×n matrix. The constant term equals det(A), so det(A) = 0 if and only if λ = 0 is an eigenvalue.
Trace relationships: the coefficient of λn−1 in det(A − λI) is (−1)n−1 tr(A) for the standard form with leading term (−1)nλn.
Explore determinants and characteristic polynomials for invariant vocabulary and singularity tests.
Singular matrices
det(A) = 0 means zero is a root of p(λ).
Invertibility
If 0 is not a root, A is invertible over ℝ (when working with real matrices).
Cofactor expansion
A practical hand method for 3×3 and 4×4 determinants with λ symbols.
Invariants
tr(A) and det(A) appear inside coefficients and eigenvalue sums/products.
Characteristic Polynomial vs Minimal Polynomial
The minimal polynomial m(λ) is the monic polynomial of least degree such that m(A) = 0. The characteristic polynomial p(λ) always satisfies p(A) = 0 by the Cayley-Hamilton theorem.
Both polynomials share the same eigenvalues, but m(λ) may use lower degree. Example: if A is a scalar multiple of I, p(λ) has degree n while m(λ) can be degree 1.
Educationally, compare them after you are comfortable computing p(λ). The minimal polynomial appears in Jordan form and advanced matrix theory.
Read characteristic vs minimal polynomial for side-by-side examples.
| Topic | Characteristic | Minimal |
|---|---|---|
| Definition | p(λ) = det(A − λI) | Monic polynomial of least degree with m(A) = 0 |
| Degree | Always n | At most n, often smaller |
| Eigenvalues | Same set as minimal | Same set as characteristic |
| Typical course level | Intro linear algebra | Second course / advanced matrix theory |
Characteristic Polynomial Calculator
The interactive tool at the top of this page is the primary calculator for this site. It stays fixed in position so you can read the guides and scroll back to the same panel.
Enter matrix entries with row labels a, b, c, d and column subscripts. Choose matrix size, pick det(A − λI) or det(λI − A), and read the simplified polynomial instantly in the browser.
The tool performs symbolic determinant expansion. It does not factor p(λ) or list eigenvalues numerically; solve p(λ) = 0 as a separate step after you have the expanded form.
For a dedicated article, see characteristic polynomial calculator guide.
- Matrix input. 2×2, 3×3, and 4×4 grids with fraction and decimal support
- Determinant variants. Switch between det(A − λI) and det(λI − A) to match your textbook
- Instant output. Expanded p(λ) updates when all cells are filled
- Privacy. All arithmetic runs locally; entries are not uploaded
Common Characteristic Polynomial Mistakes
Avoid these errors when computing p(λ) by hand or when checking calculator output.
Subtracting λ from every entry
Only diagonal entries of A change in A − λI. Off-diagonal entries stay the same.
Sign errors in expansion
Cofactor signs and det(λI − A) versus det(A − λI) flip terms. Label which definition you use.
Confusing polynomial with equation
p(λ) is the polynomial; p(λ) = 0 is the characteristic equation for eigenvalues.
Stopping at expanded form
Courses often require factoring p(λ) or solving for λ. The calculator gives p(λ) only.
Ignoring complex roots
Real matrices can have complex eigenvalues. Expect λ² + 1 when A is a 90° rotation block.
Cayley-Hamilton Theorem Explained
The Cayley-Hamilton theorem states that every square matrix satisfies its own characteristic equation: if p(λ) = det(A − λI), then p(A) = 0 (the zero matrix) when you substitute the matrix A for λ using matrix arithmetic.
For example, if p(λ) = λ² − 4λ + 3 and A is 2×2, then A² − 4A + 3I is the zero matrix. This is a powerful fact for simplifying matrix powers and proving identities.
Introductory courses may mention the theorem after eigenvalues. You do not need it to use the calculator, but it explains why characteristic polynomials are more than exam exercises.
Read Cayley-Hamilton theorem explained for a worked 2×2 check.
If p(λ) = det(A − λI), then p(A) = 0
Matrix substitute: powers A^k, matrix addition, scalar multiples