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For a degree n polynomial normalized to monic form:
The Frobenius companion matrix C is:
Its characteristic polynomial equals p(λ), so its eigenvalues are the polynomial roots.
Example polynomial: 2·λ⁴ − 3·λ³ + 0·λ² + 5·λ − 7. After normalization, divide coefficients by 2.
| Parameter | Value | Notes |
|---|---|---|
| Degree (n) | 4 | Matrix size is 4×4 |
| Leading aₙ | 2 | Will be normalized to 1 |
| a₃, a₂, a₁, a₀ | −3, 0, 5, −7 | Entered in coefficient fields |
| Normalized coefficients | a₃=−1.5, a₂=0, a₁=2.5, a₀=−3.5 | Used in the last column as negatives |
A companion matrix converts a polynomial into a state matrix whose eigenvalues match the polynomial roots. This makes it useful for control design, vibration analysis, and discrete-time models, where characteristic equations arise naturally. Because the structure is fixed, you can compare different polynomials consistently, test stability by inspecting eigenvalues, and build canonical realizations. The calculator automates the construction while keeping the mapping between coefficients and matrix entries explicit.
Most definitions assume a monic polynomial, so the leading coefficient must be nonzero and the remaining coefficients are divided by it. Normalization improves numerical conditioning and ensures the last column is simply the negative normalized coefficients. If your coefficients vary by many orders of magnitude, consider rescaling the variable, for example λ = s·μ, to reduce overflow and improve iterative root estimation. The tool shows both entered and normalized polynomials for traceability.
In the Frobenius form, the subdiagonal contains ones that shift the state, while the final column injects the coefficient information. For p(λ)=λⁿ+aₙ₋₁λⁿ⁻¹+…+a₁λ+a₀, the last column is −a₀, −a₁, …, −aₙ₋₁. Sign mistakes are common when copying formulas, so the “Quick validation” checklist helps confirm the pattern. Once verified, the matrix can be used directly in eigenvalue routines.
CSV export is ideal for spreadsheets, scripts, and numerical packages that accept comma-separated matrices. PDF export is designed for sharing a static snapshot with colleagues, assignments, or audit trails. Including both the entered and normalized polynomials in the PDF reduces ambiguity when coefficients are rounded or units are implied. For repeated studies, keep a consistent degree and use the example-fill button as a template for your own datasets.
Engineers often start with a transfer function denominator, expand it into polynomial coefficients, then build a companion matrix to obtain a state-space realization. Data scientists may use companion matrices to study linear recurrences and to compute roots as eigenvalues for forecasting. In algebra courses, the matrix provides a concrete link between polynomials and linear operators. When combined with eigenvectors, you can also form modal decompositions and sensitivity checks for parameter changes.
A companion matrix represents a monic polynomial as an n×n matrix whose characteristic polynomial is the original polynomial. It is used to study roots via eigenvalues and to build canonical state‑space models.
Normalization divides all coefficients by the leading term to make the polynomial monic. If the leading coefficient is zero, the degree changes and the construction becomes invalid.
Scaling all coefficients by the same nonzero factor does not change the normalized polynomial, so the resulting companion matrix is unchanged after normalization.
Root estimates are numerical approximations and can vary with coefficient scaling and problem conditioning. Use them as guidance, and verify critical results with a trusted numerical library when needed.
The ones shift the state basis so that powers of λ align with the state vector. This shift structure is what makes the companion form a canonical realization.
The CSV contains only the matrix values, suitable for computation. The PDF includes the entered polynomial, the normalized polynomial, and the matrix rows for clear documentation.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.