Characteristic Polynomial Solver

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Compute the characteristic polynomial \[ p(\lambda)=\det(A-\lambda I), \] show coefficients, optional numeric roots, an interactive graph of \(p(\lambda)\), and a Cayley–Hamilton check \(p(A)\approx 0\).

Matrix size

Tolerance

Display decimals

Steps

Show numeric roots (eigenvalues) Show interactive graph Show Cayley–Hamilton check \(p(A)\) Auto-scale Y on zoom

Matrix \(A\)

A is 2×2

Inputs accept -3.5, 2e-4, fractions like 7/3, and constants pi, e.

Results

Characteristic polynomial

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Coefficients

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Numeric roots (λ)

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Trace/Det relations

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Ready

Enter matrix \(A\), then click “Calculate”.

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Theory

Characteristic polynomial \(p(\lambda)=\det(A-\lambda I)\)

For a square matrix \(A\in\mathbb{R}^{n\times n}\), the characteristic polynomial is \[ p(\lambda)=\det(A-\lambda I). \] It is a degree-\(n\) polynomial whose roots are the eigenvalues of \(A\) (possibly complex).

The equation \(p(\lambda)=0\) is also called the characteristic equation. In applications, eigenvalues control stability, repeated application \(A^k\), and differential equations \(x'=Ax\).

2×2 expansion (manual-friendly)

For \(A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}\), \[ A-\lambda I=\begin{bmatrix}a-\lambda & b\\ c & d-\lambda\end{bmatrix}, \] so \[ p(\lambda)=(a-\lambda)(d-\lambda)-bc =\lambda^2-(a+d)\lambda+(ad-bc). \] This shows the classic relations: \[ \mathrm{tr}(A)=a+d,\qquad \det(A)=ad-bc. \]

3×3 coefficient relations (invariants)

For \(3\times3\) matrices, directly expanding \(\det(A-\lambda I)\) is algebra-heavy. A common approach is to compute coefficients from invariants: \[ p(\lambda)=\lambda^3-(\mathrm{tr}A)\lambda^2+c_1\lambda-\det(A), \] where \[ c_1=\frac{(\mathrm{tr}A)^2-\mathrm{tr}(A^2)}{2}. \] These quantities do not change under similarity transforms and are useful for both theory and computation.

Cayley–Hamilton theorem \(p(A)=0\)

The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic polynomial: \[ p(A)=0. \] For example, in the \(2\times2\) case: \[ A^2-(\mathrm{tr}A)A+(\det A)I=0. \] This is extremely useful to reduce high powers of \(A\) and derive recurrence relations.

Graphing \(p(\lambda)\)

When coefficients are real, the graph of \(p(\lambda)\) over real \(\lambda\) shows where the polynomial crosses zero (real eigenvalues). Complex eigenvalues occur in conjugate pairs and do not appear as x-axis crossings. The interactive graph in the calculator helps build intuition by showing how \(p(\lambda)\) grows and changes sign.

University note: companion matrices

In advanced linear algebra, characteristic polynomials connect to companion matrices and canonical forms. The companion matrix of a monic polynomial has that polynomial as its characteristic polynomial, linking roots, recurrences, and matrix representations.

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