Characteristic Polynomial Calculator
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The online Characteristic Polynomial Calculator helps you calculate instantly and solve problems related to Eigenvalues Eigenvectors. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Engineering Math calculations.
Compute the characteristic polynomial p(λ) = det(λI − A) of a square matrix. Enter Matrix A to get the monic polynomial in descending powers of λ.
p(λ) = det(λI − A)
The characteristic polynomial of an n×n matrix A is the monic degree-n polynomial p(λ) = det(λI − A). Its roots are the eigenvalues of A. For 2×2, p(λ) = λ² − tr(A)λ + det(A).
- Matrix A is square with real entries.
- Convention is p(λ) = det(λI − A), not det(A − λI).
- Coefficients are monic and listed high → low.
Inputs
Results
Worked Examples
2×2 non-diagonal matrix
Char poly of [[1,2],[3,4]] → λ² − 5λ − 2.
- matrix: [ [ 1, 2 ], [ 3, 4 ] ]
- result: λ² − 5λ − 2
- coefficients: 1, -5, -2
2×2 diagonal matrix
Char poly of diag(2,3) → λ² − 5λ + 6.
- matrix: [ [ 2, 0 ], [ 0, 3 ] ]
- result: λ² − 5λ + 6
- coefficients: 1, -5, 6
3×3 matrix
Char poly of a 3×3 matrix → monic cubic.
- matrix: [ [ 1, 2, 3 ], [ 0, 1, 4 ], [ 5, 6, 0 ] ]
- result: λ³ − 2λ² − 38λ − 1
- coefficients: 1, -2, -38, -1
About this calculator
Overview
The characteristic polynomial of a square matrix A is p(λ) = det(λI − A). It is monic of degree n for an n×n matrix, and its roots are exactly the eigenvalues of A.
When to use
Use this calculator to find p(λ) before computing eigenvalues, to check Cayley–Hamilton exercises, or to verify a hand-derived det(λI − A) expansion.
Inputs explained
- Matrix A — square n×n numeric matrix (2×2 and 3×3 emphasized; up to 8×8 supported)
Formula / method
Form λI − A and compute its determinant as a polynomial in λ. Coefficients are returned monic, high→low. This page uses det(λI − A), which differs from det(A − λI) by a factor of (−1)^n.
Worked example
Example: A = [[1, 2], [3, 4]] gives p(λ) = λ² − 5λ − 2, coefficients [1, −5, −2].
Interpreting results
The primary result is the pretty polynomial p(λ). Coefficients list the same polynomial in descending powers. Degree equals the matrix order.
Assumptions
- A is square with real entries.
- Convention is consistently p(λ) = det(λI − A).
Limitations
- Not a full computer-algebra system for symbolic entries.
- Floating-point coefficients may be rounded for display.
- Critical engineering decisions should be cross-checked by hand or with certified software.
Important note
Educational engineering-mathematics tool. Verify critical designs with hand analysis or certified software.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
Form λI − A and compute its determinant as a polynomial in λ. Coefficients are returned monic, high→low. This page uses det(λI − A), which differs from det(A − λI) by a factor of (−1)^n.
Example Calculation
Example: A = [[1, 2], [3, 4]] gives p(λ) = λ² − 5λ − 2, coefficients [1, −5, −2].
Frequently Asked Questions
What is Characteristic Polynomial Calculator?
- The characteristic polynomial of a square matrix A is p(λ) = det(λI − A). It is monic of degree n for an n×n matrix, and its roots are exactly the eigenvalues of A.
How does Characteristic Polynomial Calculator work?
- Form λI − A and compute its determinant as a polynomial in λ. Coefficients are returned monic, high→low. This page uses det(λI − A), which differs from det(A − λI) by a factor of (−1)^n.
Why use this Engineering Math calculator?
- Use this calculator to find p(λ) before computing eigenvalues, to check Cayley–Hamilton exercises, or to verify a hand-derived det(λI − A) expansion.