Characteristic Polynomial Calculator
Definition and Intuition
For a square matrix A, the characteristic polynomial is a single polynomial that captures key information about how the matrix acts on vectors. It is defined as the determinant of A minus a scalar multiple of the identity matrix:
p(λ) = det(A − λI), where λ (lambda) is a scalar variable and I is the identity matrix of the same size as A.
Conceptually, this polynomial measures when the matrix A − λI becomes singular (non-invertible). Values of λ that make the determinant zero are the
eigenvalues of A. These values describe how the matrix stretches, compresses, rotates, or reflects space along special directions called eigenvectors.
For an n × n matrix, the characteristic polynomial has degree n and can be written in the general form
p(λ) = λⁿ + c₁ λⁿ⁻¹ + c₂ λⁿ⁻² + … + cn−1 λ + cn,
where the coefficients ck are built from invariants of the matrix such as the trace and determinant.
Formal Definition in MathML
The definition can also be written using MathML. For an n × n matrix A, the characteristic polynomial is
Here, is a scalar variable, and is the identity matrix.
Connection to Eigenvalues
The roots of the characteristic polynomial are precisely the eigenvalues of the matrix. If
p(λ) = 0 for some value λ = λ₀, then there exists a nonzero vector v such that
A v = λ₀ v. This vector v is an eigenvector associated with the eigenvalue λ₀.
In applications, eigenvalues describe important qualitative features:
- In systems of differential equations, they indicate whether solutions grow, decay, or oscillate.
- In vibration and structural analysis, they correspond to natural frequencies and modes.
- In Markov chains, they are related to long-term behavior and convergence to steady states.
The characteristic polynomial is therefore a compact way to encode this eigenvalue information without computing eigenvectors directly.
Formulas for 2×2, 3×3, and 4×4 Matrices
2×2 matrices
For a 2×2 matrix
A = [[a, b], [c, d]],
the characteristic polynomial is
p(λ) = λ² − (tr A) λ + det(A),
where
tr A = a + dis the trace (sum of diagonal entries), anddet(A) = ad − bcis the determinant.
3×3 matrices
For a 3×3 matrix, the characteristic polynomial has degree three. It can be written as
p(λ) = λ³ − (tr A) λ² + s λ − det(A),
where s is a symmetric expression in the entries of A (related to the sum of principal 2×2 minors or the trace of A²). The calculator uses a stable
algorithm to compute this coefficient directly from the matrix without requiring you to expand determinants by hand.
4×4 matrices
For a 4×4 matrix, the characteristic polynomial has degree four. The general pattern continues, and the coefficients can be expressed using traces of powers of A and
combinations of minors, but the explicit formulas become quite long. Instead of asking you to apply those formulas manually, this calculator uses the Faddeev–Leverrier algorithm to
generate them from the matrix entries.
How This Calculator Works
This page supports real-valued 2×2, 3×3, and 4×4 matrices. You select the matrix size, enter the entries, and then the tool computes the characteristic polynomial
p(λ) in descending powers of λ.
Internally, the implementation uses the Faddeev–Leverrier algorithm. Very roughly, this algorithm:
- Starts from the original matrix
A. - Builds a sequence of matrices using matrix multiplication and additions with scaled identity matrices.
- Extracts each new coefficient of the characteristic polynomial from the trace of the current matrix in the sequence.
Because it works with traces and matrix products, this approach avoids large symbolic determinant expansions and scales well to 4×4 matrices.
Interpreting the Results
After you click Compute Polynomial, the tool returns an expression such as
p(λ) = λ³ − 12 λ² + 47 λ − 60.
You can read this as an ordinary polynomial in the variable λ. Key points when interpreting it include:
-
Degree: The degree (the highest power of
λ) equals the dimension of the matrix. A 2×2 matrix gives a quadratic, a 3×3 gives a cubic, and a 4×4 gives a quartic. -
Leading coefficient: For matrices in this tool, the coefficient of the highest power of
λis 1, so the polynomial is monic. -
Trace and determinant: For 2×2 and 3×3 matrices, the coefficient of
λⁿ⁻¹is (up to a sign) the trace, and the constant term (up to a sign) is the determinant. - Repeated roots: If the polynomial has repeated roots, the matrix has repeated eigenvalues. The matrix may or may not be diagonalizable in that case; you would need eigenvectors to decide.
- Complex eigenvalues: Even though you enter real numbers, the roots of the polynomial can be complex. The calculator focuses on the polynomial itself, which you can then use in a root-finding tool or algebra system.
The characteristic polynomial alone tells you the eigenvalues (via its roots) but not the eigenvectors. If you need eigenvectors, you will typically combine this result with an eigenvalue or eigenvector calculator.
Worked 3×3 Example
Consider the 3×3 matrix
A = [[4, 2, 0], [1, 3, 0], [0, 0, 5]].
Step 1: Compute the trace and determinant.
- Trace:
tr A = 4 + 3 + 5 = 12. -
Determinant: because the matrix is block upper triangular, the determinant is the product of the diagonal entries:
det(A) = 4 × 3 × 5 = 60.
Step 2: Form A − λI. This gives
A − λI = [[4 − λ, 2, 0], [1, 3 − λ, 0], [0, 0, 5 − λ]].
Step 3: Compute the determinant det(A − λI). The lower-right block is 1×1, so
det(A − λI) = (5 − λ) det([[4 − λ, 2], [1, 3 − λ]]).
The 2×2 determinant is
(4 − λ)(3 − λ) − 2 × 1 = (12 − 7λ + λ²) − 2 = 10 − 7λ + λ².
Therefore,
p(λ) = det(A − λI) = (5 − λ)(10 − 7λ + λ²).
Expanding and simplifying yields
p(λ) = λ³ − 12 λ² + 47 λ − 60.
The roots of this cubic are λ = 5, λ = 4, and λ = 3, which are exactly the diagonal entries in this example. For a non-triangular matrix, the roots will
generally not be visible from the entries, and a calculator like this is especially useful.
Comparison: 2×2 vs 3×3 vs 4×4
The table below summarizes how the characteristic polynomial behaves for the different matrix sizes supported here.
| Matrix size | Polynomial degree | Typical form of p(λ) | Trace / determinant relationship | Typical use cases |
|---|---|---|---|---|
| 2×2 | 2 (quadratic) | λ² − (tr A) λ + det(A) |
Coefficient of λ is − tr A; constant term is det(A). |
Intro linear algebra, simple dynamical systems, 2-state Markov chains. |
| 3×3 | 3 (cubic) | λ³ − (tr A) λ² + s λ − det(A) |
Coefficient of λ² is − tr A; constant term is − det(A); middle coefficient involves traces and 2×2 minors. |
Three-dimensional systems, vibration modes, 3-state Markov chains. |
| 4×4 | 4 (quartic) | λ⁴ + c₁ λ³ + c₂ λ² + c₃ λ + c₄ |
Coefficients c₁, …, c₄ are combinations of traces and minors, computed algorithmically. |
Small control systems, 4-variable models, compact test cases in numerical linear algebra. |
How to Use the Calculator
- Select the matrix size (2×2, 3×3, or 4×4) from the Matrix Size menu.
- Enter each matrix entry as a decimal number. You can use negative numbers and decimals like
-1.5or2.75. - Check that all fields are filled for the chosen size. If you change the dimension, the input grid will adjust.
- Click Compute Polynomial to generate
p(λ). - Optionally use the Copy Result button to paste the polynomial into your notes, reports, or another tool (for example, a root finder or eigenvalue calculator).
The result is displayed in descending powers of λ with coefficients simplified numerically. If any required input is missing or invalid, the calculator will display an error so
you can correct it and try again.
Assumptions and Limitations
To keep the tool fast and focused, it makes several assumptions:
- Supported sizes: Only 2×2, 3×3, and 4×4 matrices are supported.
- Input type: Entries are treated as real-valued decimal numbers. Complex inputs are not supported directly.
- Numeric precision: Coefficients are computed using floating-point arithmetic. Very large or very small numbers, or highly ill-conditioned matrices, may lead to numerically sensitive coefficients.
- Output format: The calculator focuses on the characteristic polynomial. It does not factor the polynomial or return eigenvectors.
- Rounding: For display, coefficients may be rounded to a reasonable number of decimal places. Internally, computations can use more precision than is shown.
If you need exact symbolic expressions (for example, with square roots or rational coefficients), you may want to use a computer algebra system in addition to this numeric calculator.
Who Uses This Calculator?
This tool is designed for:
- Students in linear algebra or differential equations courses who need quick feedback when computing characteristic polynomials by hand.
- Engineers performing stability analysis on small linear systems or control models.
- Data scientists and analysts who want to check basic eigenvalue-related properties of compact matrices before moving to larger numerical routines.
By automating the most error-prone algebra, the calculator lets you focus on interpreting the eigenvalues and understanding what they mean for your specific problem.