Characteristic Polynomial 4x4 Matrix Calculator

Enter 4x4 Matrix Values

Use decimals, whole numbers, negative values, or zero. The calculator fills a sample matrix by default.

a11

a12

a13

a14

a21

a22

a23

a24

a31

a32

a33

a34

a41

a42

a43

a44

Advanced Options

Decimal precision

Zero tolerance

Graph points

Graph λ minimum

Graph λ maximum

Polynomial convention

Formula Used

Characteristic polynomial:

p(λ) = det(λI − A)

p(λ) = λ⁴ + c₁λ³ + c₂λ² + c₃λ + c₄

For a 4x4 matrix, c₄ equals det(A) under this convention.

The tool uses the Faddeev LeVerrier method. Start with B₀ = I. Then compute cₖ = −tr(ABₖ₋₁) / k. Next compute Bₖ = ABₖ₋₁ + cₖI. Repeat for k = 1, 2, 3, and 4.

How To Use This Calculator

Enter all sixteen values of the 4x4 matrix.
Set decimal precision and zero tolerance if needed.
Choose the λ range and graph point count.
Press the calculate button.
Read the polynomial, coefficients, trace steps, and graph.
Download the CSV or PDF file for later use.

Example Data Table

Matrix Type	Matrix	Expected Polynomial
Diagonal	diag(1, 2, 3, 4)	λ⁴ − 10λ³ + 35λ² − 50λ + 24
Scalar identity	2I	λ⁴ − 8λ³ + 24λ² − 32λ + 16
Nilpotent shift	Superdiagonal ones, zeros elsewhere	λ⁴

What This Calculator Does

A characteristic polynomial turns a square matrix into one polynomial. For a 4x4 matrix, the answer has degree four. Its roots are the eigenvalues of the matrix. Those values explain stretching, rotation, stability, vibration, and many other behaviors. This calculator focuses on det(lambda I minus A). That convention gives a leading coefficient of one.

Why The Polynomial Matters

A 4x4 matrix can describe four linked variables. In algebra, physics, control systems, and data models, the characteristic polynomial gives a compact summary of that link. The coefficient of lambda cubed is the negative trace. The constant term is the determinant. Middle coefficients include deeper trace relationships. These checks help you spot input errors.

How The Calculation Works

The tool uses the Faddeev LeVerrier method. It avoids writing the full determinant expansion by hand. The method builds coefficients from repeated matrix products and traces. It is efficient for a 4x4 matrix. It also gives useful intermediate checks. The final form is lambda to the fourth power plus four coefficient terms.

Good Input Practice

Enter each matrix value carefully. Decimals and negative numbers are allowed. Use zero when a position is empty. Choose enough precision for your task. A small zero tolerance can hide harmless rounding noise. A larger tolerance can simplify printed results, but it may also hide small real effects.

Reading The Results

Start with the polynomial line. Then review the coefficient table. Check the trace and determinant fields. The Cayley Hamilton residual should be close to zero for normal numeric inputs. The graph shows how the polynomial changes across a chosen lambda range. Crossings near the horizontal axis suggest real eigenvalue locations.

Using Exports

The CSV file is useful for spreadsheets. The PDF file is useful for records, lessons, and reports. Save both when you need to compare matrices. You can also copy the polynomial into another algebra tool. Always keep the original matrix with the result. That makes later checking much easier.

Common Uses

Students use it for eigenvalue practice. Engineers use it for stability checks. Analysts use it when studying transformations. Teachers use it to create examples. The matrix can be tested again.

FAQs

What is a characteristic polynomial?

It is the polynomial made from det(λI − A). For a 4x4 matrix, it has degree four. Its roots are the matrix eigenvalues.

Does this calculator support decimal values?

Yes. You can enter integers, decimals, negative values, and zeros. The precision setting controls how many decimal places appear in the final result.

Which polynomial convention is used?

The calculator uses p(λ) = det(λI − A). For a 4x4 matrix, det(A − λI) gives the same polynomial because the size is even.

Does it calculate eigenvalues?

It calculates the characteristic polynomial. Eigenvalues are the roots of that polynomial. The graph can help you visually locate real roots.

What is zero tolerance?

Zero tolerance treats very small values as zero. It helps clean results caused by floating point rounding. Avoid making it too large.

What is the Cayley Hamilton residual?

It checks whether p(A) is close to the zero matrix. A tiny residual means the computed polynomial is internally consistent.

Why is the determinant shown?

The determinant is the constant term for this 4x4 convention. It is useful for checking invertibility and validating the polynomial.

Can I save the result?

Yes. Use the CSV button for spreadsheet data. Use the PDF button for a readable report containing the main calculated values.