Characteristic Polynomial Calculator
Our free linear algebra calculator solves characteristic polynomial problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Characteristic Polynomial Calculator
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Formula: p(x) = det(A - xI) = x^2 - tr(A)x + det(A)
Worked example — p(x) = x^2 - 7x + 10 | Eigenvalues: 5 and 2
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 Determinant = 4*3 - 1*2 = 10 Characteristic polynomial: p(x) = x^2 - 7x + 10 Discriminant: 49 - 40 = 9 > 0 (real distinct eigenvalues) Eigenvalues: x = (7 +/- sqrt(9)) / 2 = (7 +/- 3) / 2 lambda_1 = 5, lambda_2 = 2 Verification: 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 Determinant = 1*1 - (-2)*3 = 1 + 6 = 7 Characteristic polynomial: p(x) = x^2 - 2x + 7 Discriminant: 4 - 28 = -24 < 0 (complex eigenvalues) Eigenvalues: x = (2 +/- sqrt(-24)) / 2 = 1 +/- i*sqrt(6) lambda_1 = 1 + 2.449i, lambda_2 = 1 - 2.449i Modulus = 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 is the Cayley-Hamilton Theorem?
The Cayley-Hamilton Theorem states that every square matrix satisfies its own characteristic polynomial. If the characteristic polynomial of matrix A is p(x) = x^2 - tr(A)x + det(A), then p(A) = A^2 - tr(A)*A + det(A)*I = 0 (the zero matrix). This remarkable theorem means you can express any power of A higher than n-1 as a linear combination of I, A, A^2, ..., A^(n-1). This is useful for computing matrix exponentials, solving differential equations, and simplifying matrix expressions. The theorem holds for all square matrices, regardless of whether they are diagonalizable or have complex eigenvalues.
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.
How are the trace and determinant related to eigenvalues?
For any square matrix, the trace equals the sum of all eigenvalues, and the determinant equals the product of all eigenvalues. For a 2x2 matrix with eigenvalues lambda_1 and lambda_2: tr(A) = lambda_1 + lambda_2 and det(A) = lambda_1 * lambda_2. These relationships follow directly from Vietas formulas applied to the characteristic polynomial. This connection is powerful because trace and determinant are easy to compute (O(n) and O(n^3) respectively), providing immediate information about eigenvalues without fully solving the characteristic equation. For example, if det(A) = 0, at least one eigenvalue must be zero.
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.
Can two different matrices have the same characteristic polynomial?
Yes, matrices with the same characteristic polynomial are called isospectral matrices, meaning they share the same eigenvalues with the same multiplicities. However, they may not be similar (related by a change of basis). For example, the 2x2 identity matrix and any matrix with trace 2 and determinant 1 share the same characteristic polynomial p(x) = x^2 - 2x + 1, but they could have different eigenvector structures. The Jordan normal form provides additional information beyond the characteristic polynomial that fully classifies matrices up to similarity. The minimum polynomial, which divides the characteristic polynomial, helps distinguish between isospectral matrices.
How does the characteristic polynomial generalize to larger matrices?
For an n x n matrix, the characteristic polynomial is a degree-n polynomial whose coefficients are related to the principal minors of the matrix. The coefficient of x^(n-1) is always -tr(A), the constant term is (-1)^n * det(A), and intermediate coefficients involve sums of principal minors of various sizes. Computing the characteristic polynomial of large matrices is done using methods like the Faddeev-LeVerrier algorithm, which recursively computes the coefficients in O(n^4) operations, or the more efficient Berkowitz algorithm. For sparse matrices, iterative methods can find individual eigenvalues without explicitly forming the characteristic polynomial.
What is the minimum polynomial and how does it relate to the characteristic polynomial?
The minimum polynomial of a matrix is the polynomial of lowest degree that the matrix satisfies (analogous to the Cayley-Hamilton theorem, but minimal). The minimum polynomial always divides the characteristic polynomial, and both share the same roots (eigenvalues), but the minimum polynomial may have lower multiplicities. For a diagonalizable matrix, the minimum polynomial has no repeated roots regardless of eigenvalue multiplicity in the characteristic polynomial. The minimum polynomial determines the structure of the Jordan normal form and is essential for computing matrix functions. Finding the minimum polynomial requires checking successive powers of the matrix for linear dependence.
How is the characteristic polynomial used in graph theory?
In graph theory, the characteristic polynomial of the adjacency matrix (called the graph characteristic polynomial) encodes important structural information about the graph. Graphs with the same characteristic polynomial are called cospectral graphs. The eigenvalues of the adjacency matrix reveal properties like the number of edges (related to the sum of squared eigenvalues), the number of triangles (related to the sum of cubed eigenvalues), and bounds on the chromatic number. The largest eigenvalue (spectral radius) bounds the maximum degree and is related to graph expansion properties. Spectral graph theory uses these connections extensively in network analysis, community detection, and graph classification algorithms.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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