Characteristic Polynomial of a Matrix Calculator

Example data table

Formula used

The characteristic polynomial of a square matrix A is defined by p(λ) = det(λI - A).

This calculator uses the Faddeev-LeVerrier process to generate the coefficients of the monic polynomial directly from matrix traces and matrix products.

If p(λ) = λⁿ + c₁λ^n-1 + ... + c_n, then c₁ = -tr(A), and the constant term equals (-1)ⁿdet(A).

These identities help verify the output quickly. The graph then evaluates the real polynomial p(x) over your selected x-range.

How to use this calculator

1. Select the matrix order from 1 to 6.
2. Enter matrix values in the grid, or paste rows in the quick paste area.
3. Set the graph minimum, maximum, and step size.
4. Choose the number of decimal places for displayed values.
5. Press the calculate button to show the result above the form.
6. Review the polynomial, coefficient table, trace, determinant, and graph.
7. Use the CSV button for tabular export.
8. Use the PDF button for a clean summary document.

Understanding the characteristic polynomial

Why this polynomial matters

The characteristic polynomial is one of the most useful summaries of a square matrix. It connects matrix structure to eigenvalues, determinant, trace, invertibility, and stability checks. When you solve p(λ) = 0, the roots are the eigenvalues of the matrix. That link makes the polynomial important in algebra, differential equations, optimization, control systems, and numerical analysis.

What the coefficients reveal

The leading coefficient is always 1 for p(λ) = det(λI - A). The second coefficient is the negative trace, so it measures the sum of diagonal entries immediately. The constant term equals (-1)ⁿdet(A), which means it also reveals whether the matrix is singular. If the determinant is zero, then the constant term is zero and the matrix has 0 as an eigenvalue.

Why the graph still helps

The graph does not show complex roots directly, but it still gives strong insight. Real x-intercepts correspond to real eigenvalues. Turning points show where the polynomial changes growth behavior. A wide graph range can expose sign changes, repeated roots, or fast divergence for high-degree matrices. This is especially useful when you want a quick visual check before deeper symbolic or numeric work.

How this page computes the result

This page uses a stable coefficient-building process rather than expanding a determinant symbol by symbol. That approach keeps the tool practical for repeated use and helps you inspect coefficients, trace, determinant, and graph values in one place. The export tools make it easier to document results for homework, notes, engineering checks, or classroom examples.

FAQs

1. What is a characteristic polynomial?

It is the polynomial p(λ) = det(λI - A) built from a square matrix A. Its roots are the matrix eigenvalues.

2. Why must the matrix be square?

Only square matrices have determinants of λI - A. That determinant is required to define the characteristic polynomial correctly.

3. What does the constant term mean?

The constant term equals (-1)ⁿdet(A). It links the polynomial directly to matrix invertibility and the presence of zero eigenvalues.

4. Does the graph show all eigenvalues?

No. The graph shows real polynomial values only. Real roots appear as x-intercepts, but complex eigenvalues do not appear as visible crossings.

5. Why is the trace shown separately?

The trace is a quick verification value. For p(λ) = λⁿ + c₁λ^n-1 + ..., the coefficient c₁ equals -tr(A).

6. What happens if the determinant is zero?

The matrix is singular, and zero is an eigenvalue. In that case, the constant term of the characteristic polynomial becomes zero.

7. Can I paste decimals or negative values?

Yes. The quick paste area and the input grid accept integers, decimals, and negative entries for valid square matrices.

8. What is the best graph range to use?

Start with a moderate range such as -10 to 10. Then widen or tighten the interval if you want a clearer view of real roots.