Formula Used
The standard characteristic polynomial is p(λ) = det(λI - A).
For an n by n matrix, p(λ) = λ^n + c1λ^(n-1) + c2λ^(n-2) + ... + cn.
This calculator uses the Faddeev LeVerrier recurrence: ck = -(1/k) × sum from i = 1 to k of c(k-i) trace(A^i), with c0 = 1.
The trace is the sum of diagonal entries. The determinant is checked separately with numeric elimination.
How to Use This Calculator
- Select the matrix size.
- Enter each matrix value in the visible square grid.
- Choose the determinant convention.
- Set decimal places for the displayed result.
- Enter a λ value when you want polynomial evaluation.
- Press Calculate, or export the completed report.
Example Data Table
| Matrix | Trace | Determinant | Characteristic Polynomial |
|---|---|---|---|
| [[4, 1], [2, 3]] | 7 | 10 | λ^2 - 7λ + 10 |
| [[1, 0], [0, 5]] | 6 | 5 | λ^2 - 6λ + 5 |
| [[2, 1, 0], [0, 3, 4], [1, 0, 2]] | 7 | 16 | λ^3 - 7λ^2 + 16λ - 16 |
Understanding Characteristic Polynomials
A characteristic polynomial turns a square matrix into one polynomial. It describes deep structure in a compact way. The roots of the polynomial are eigenvalues. Those values control stretching, rotation, stability, and many linear transformations. This calculator helps you build the polynomial without losing track of signs.
Why It Matters
Matrices appear in systems of equations, computer graphics, statistics, physics, and engineering. A characteristic polynomial gives fast clues about the behavior of a matrix. Its coefficients include important invariants. The first coefficient after the leading term is linked to the trace. The constant term is linked to the determinant. These facts make the result useful for checking work.
What The Calculator Does
Enter any square matrix from size two to five. The tool computes powers of the matrix. It then applies the Faddeev LeVerrier method. This method uses traces of powers to create each coefficient. It avoids expanding a large determinant by hand. That keeps the process organized and repeatable.
You can choose the standard determinant form det(lambda I minus A). You can also view the alternate form det(A minus lambda I). Both forms contain the same eigenvalue information. Their signs may differ when the matrix order is odd. The calculator also evaluates the polynomial at a chosen value of lambda.
Reading The Output
The result shows the polynomial, coefficient list, trace, determinant, power traces, and a Cayley Hamilton check. The Cayley Hamilton theorem says a square matrix satisfies its own polynomial. A small residual means the numeric computation is consistent. Decimal controls help you present clean answers.
Good Input Practice
Use exact integers when possible. Use decimals only when the matrix comes from measurements. Keep rows and columns in the correct order. Review every entry before calculating. For classroom problems, compare the coefficient of the next highest power with the negative trace. Compare the constant term with the determinant sign rule. These two checks catch many mistakes.
Exports And Study Use
CSV export is useful for spreadsheets. PDF export is useful for notes, worksheets, and reports. Save both when you need to document steps. The example table shows typical matrices and expected patterns. Use it to test the page before entering your own matrix.
FAQs
What is a characteristic polynomial?
It is the polynomial formed from det(λI - A). Its roots are the eigenvalues of the square matrix.
Which matrix sizes are supported?
This page supports square matrices from 2 x 2 through 5 x 5. Larger sizes need more computation and display space.
Why does the alternate convention change signs?
det(A - λI) equals det(λI - A) multiplied by (-1)^n. Odd matrix orders therefore flip every coefficient sign.
Can decimals be used?
Yes. Decimal entries work well for measured data. Use more decimal places when small coefficient differences matter.
What does the Cayley Hamilton residual mean?
It measures how close p(A) is to the zero matrix. A very small value supports the computed polynomial.
Are eigenvalues always shown?
Direct roots are shown for 2 x 2 matrices. Higher degree exact roots can be complicated, so the polynomial is reported.
What is the fastest manual check?
Compare the second coefficient with the negative trace. Then compare the constant term with the determinant sign rule.
What exports are available?
You can download a CSV report for spreadsheet work and a PDF report for notes or class records.