Characteristic Polynomial Calculator
Our free linear algebra calculator solves characteristic polynomial problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
p(x) = det(A - xI) = x^2 - tr(A)x + det(A)
For a 2x2 matrix A, the characteristic polynomial is p(x) = x^2 - tr(A)*x + det(A), where tr(A) is the trace (sum of diagonal entries) and det(A) is the determinant. The eigenvalues are the roots of p(x) = 0.
Worked Examples
Example 1: 2x2 Matrix with Real Eigenvalues
Problem:Find the characteristic polynomial and eigenvalues of A = [[4, 1], [2, 3]].
Solution:Trace = 4 + 3 = 7\nDeterminant = 4*3 - 1*2 = 10\nCharacteristic polynomial: p(x) = x^2 - 7x + 10\nDiscriminant: 49 - 40 = 9 > 0 (real distinct eigenvalues)\nEigenvalues: x = (7 +/- sqrt(9)) / 2 = (7 +/- 3) / 2\nlambda_1 = 5, lambda_2 = 2\nVerification: 5 + 2 = 7 = trace, 5 * 2 = 10 = det
Result:p(x) = x^2 - 7x + 10 | Eigenvalues: 5 and 2
Example 2: 2x2 Matrix with Complex Eigenvalues
Problem:Find the characteristic polynomial and eigenvalues of A = [[1, -2], [3, 1]].
Solution:Trace = 1 + 1 = 2\nDeterminant = 1*1 - (-2)*3 = 1 + 6 = 7\nCharacteristic polynomial: p(x) = x^2 - 2x + 7\nDiscriminant: 4 - 28 = -24 < 0 (complex eigenvalues)\nEigenvalues: x = (2 +/- sqrt(-24)) / 2 = 1 +/- i*sqrt(6)\nlambda_1 = 1 + 2.449i, lambda_2 = 1 - 2.449i\nModulus = sqrt(1 + 6) = sqrt(7) = 2.646
Result:p(x) = x^2 - 2x + 7 | Eigenvalues: 1 + 2.449i and 1 - 2.449i
Frequently Asked Questions
What is a characteristic polynomial of a matrix?
The characteristic polynomial of a square matrix A is defined as p(x) = det(A - xI), where I is the identity matrix and x (often written as lambda) is a variable. For a 2x2 matrix, this produces a quadratic polynomial: p(x) = x^2 - tr(A)x + det(A), where tr(A) is the trace (sum of diagonal elements) and det(A) is the determinant. For an n x n matrix, the characteristic polynomial is a degree-n polynomial. The roots of the characteristic polynomial are the eigenvalues of the matrix, making it one of the most important polynomials in linear algebra and matrix theory.
How do you find eigenvalues from the characteristic polynomial?
Eigenvalues are found by setting the characteristic polynomial equal to zero and solving for x. For a 2x2 matrix with characteristic polynomial p(x) = x^2 - tr(A)x + det(A) = 0, apply the quadratic formula: x = (tr(A) plus or minus sqrt(tr(A)^2 - 4*det(A))) / 2. The discriminant D = tr(A)^2 - 4*det(A) determines the nature of eigenvalues. If D > 0, there are two distinct real eigenvalues. If D = 0, there is one repeated real eigenvalue. If D < 0, the eigenvalues are complex conjugates. For larger matrices, numerical methods like the QR algorithm are used since polynomials of degree 5 or higher have no general closed-form solution.
What does the discriminant of the characteristic polynomial tell us?
The discriminant D = tr(A)^2 - 4*det(A) of a 2x2 characteristic polynomial reveals the geometric nature of the linear transformation. When D > 0, the matrix has two distinct real eigenvalues, meaning it stretches space differently in two independent directions. When D = 0, the repeated eigenvalue indicates the matrix might be a scaling (if diagonalizable) or has a Jordan block structure. When D < 0, the complex eigenvalues indicate the transformation involves rotation combined with scaling. In dynamical systems, the sign of the discriminant determines whether trajectories are nodes, spirals, or degenerate cases.
What is the significance of the characteristic polynomial in stability analysis?
In control theory and dynamical systems, the characteristic polynomial determines system stability. For a linear system dx/dt = Ax, the system is asymptotically stable if and only if all roots (eigenvalues) of the characteristic polynomial have negative real parts. For discrete systems x(k+1) = Ax(k), stability requires all eigenvalues to have magnitude less than 1. The Routh-Hurwitz criterion provides a method to determine stability directly from the characteristic polynomial coefficients without computing eigenvalues. This makes the characteristic polynomial the central object in classical control theory, feedback design, and stability certification.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy