Formula used
The characteristic polynomial of a square matrix A is: p(λ) = det(λI − A). It expands as λn + c1λn−1 + … + cn.
This calculator computes coefficients using the Faddeev–LeVerrier recursion: start with B0=I, then for k=1…n: Bk=A·Bk−1, ck=−(1/k)·tr(Bk), and update Bk←Bk+ckI.
How to use this calculator
- Select the matrix size n×n.
- Enter values in the grid, or paste CSV rows.
- Pick the variable symbol and rounding precision.
- Optionally enable the step table for learning.
- Press compute; results appear above the form.
- Use CSV or PDF buttons to save the report.
Example data table
| Size | Matrix A | Characteristic polynomial p(λ) |
|---|---|---|
| 2×2 | [ [2, 1], [1, 2] ] | λ2 − 4λ + 3 |
| 3×3 | diag(1, 2, 3) | λ3 − 6λ2 + 11λ − 6 |
| 3×3 | [ [0,1,0], [0,0,1], [0,0,0] ] | λ3 |
Coefficient meaning and invariants
The characteristic polynomial p(λ)=det(λI−A) compresses key matrix invariants into one expression. For an n×n matrix, the calculator returns n coefficients c₁…cₙ, where c₁=−tr(A) and cₙ=(−1)ⁿdet(A). These identities provide quick quality checks and help connect trace, determinant, and eigenstructure. They summarize eigenvalue combinations without computing eigenvalues.
Workflow and input validation
You can enter values in the grid or paste a comma‑separated matrix with one row per line. The tool enforces square sizes from 2×2 up to 8×8 to keep computations responsive. Each entry is validated as a finite number before any algebra starts, reducing silent failures caused by blanks, text, or overflow. When CSV mode is enabled, the grid is hidden to prevent mixed inputs.
Performance notes for practical sizes
Coefficients are computed with the Faddeev–LeVerrier recursion, which relies on matrix products and traces instead of symbolic expansion. With straightforward multiplication, runtime scales roughly with O(n⁴), so limiting n to 8 keeps calculations fast on typical devices. For many teaching and analysis tasks, 2≤n≤6 delivers instant feedback while still covering realistic examples. Memory use stays modest because only a few working matrices are stored per step, even for dense matrices.
Interpreting steps and checks
If you enable steps, the table reports trace(A·Bₖ₋₁) and the derived coefficient cₖ=−trace/k. Small rounding differences are normal, especially when eigenvalues are close or entries vary greatly in magnitude. The additional determinant comparison highlights when numeric noise is large enough to affect the constant term. If the check note appears, increasing precision or using simpler integers often clarifies the result. For sensitive matrices, consider comparing outputs across two precisions to gauge stability.
Practical applications and reporting
Characteristic polynomials support eigenvalue computation, stability studies in linear systems, diagonalization checks, and minimal‑polynomial exploration. In control and differential‑equation work, the polynomial’s roots indicate growth or decay behavior. For data workflows, the included CSV export is convenient for spreadsheets, while the PDF report is useful for assignments and documentation. Pair the exported coefficients with your preferred solver to approximate eigenvalues when needed.
FAQs
1) What does this calculator output?
It outputs the expanded characteristic polynomial, a coefficient table, optional step traces, and a determinant-based sanity check. You can also download CSV and PDF summaries.
2) Why is the constant term tied to det(A)?
Because p(0)=det(−A)=(−1)ⁿdet(A). In expanded form, p(0)=cₙ, so cₙ should match up to rounding.
3) Can it find eigenvalues directly?
It builds the polynomial and coefficients. To estimate eigenvalues, export coefficients and use a numeric root-finder or an eigenvalue routine in your preferred tool.
4) How should I format pasted CSV input?
Paste exactly n lines, each with n comma-separated numbers. Avoid headers and extra commas. Example for 2×2: 2,1 then 1,2.
5) What precision should I choose?
Start with 6 digits for readability. Increase precision when coefficients are small or nearly cancel. Decrease precision for clean integers when your matrix entries are simple.
6) Why might results differ from another tool?
Different algorithms and rounding choices can shift tiny digits. Near-singular matrices and large value ranges amplify floating-point error. Use the step table and determinant check to judge sensitivity.