Eigenvalue Preview from Characteristic Polynomial

Math Linear Algebra • Determinants and Rank

Written by STEM Calculators Team Published February 22, 2026 Updated February 24, 2026

Compute the characteristic polynomial \(p(\lambda)=\det(A-\lambda I)\) and preview eigenvalues as its roots. You can also switch to the monic convention \(p(\lambda)=\det(\lambda I-A)\) (same roots). Includes polynomial expansion, a numeric complex root finder, and a real-axis graph to visualize real roots.

Size (n×n)

Characteristic polynomial

Root mode

Display decimals

Tolerance

Graph range

Steps

Show complex eigenvalues Show stability preview (Re(λ) sign)

Entries accept -3.5, 2e-4, fractions like 7/3, and constants pi, e. The graph shows the real polynomial \(p(x)\) with numbered axes. Real eigenvalues appear where the curve crosses \(y=0\).

Matrix \(A\)

2×2

Results

Characteristic polynomial

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Coefficients (ascending)

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Eigenvalues (roots)

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Root check \(|p(\lambda_k)|\)

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Stability preview

Re(λ) < 0

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Re(λ) ≈ 0

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Re(λ) > 0

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Interpretation

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Ready

Polynomial graph on the real axis

Plots \(y=p(x)\) for real \(x\) with numbered ticks. Real eigenvalues appear where the curve crosses \(y=0\).

Drag to pan • wheel to zoom • double-click to reset view • Play animates a sweep over x

Enter a matrix and click “Calculate”.

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Theory

Characteristic polynomial and eigenvalues

For a square matrix \(A\in\mathbb{R}^{n\times n}\), the characteristic polynomial is built from a determinant that includes a parameter \(\lambda\): \[ p(\lambda)=\det(A-\lambda I). \] A number \(\lambda\) is an eigenvalue of \(A\) exactly when it makes the matrix \(A-\lambda I\) singular, meaning \(\det(A-\lambda I)=0\). Therefore, eigenvalues are precisely the roots of \(p(\lambda)\).

Many textbooks instead define the monic polynomial \(p(\lambda)=\det(\lambda I-A)\), whose leading coefficient is \(+1\). The two conventions differ only by an overall sign factor \((−1)^n\), so they have the same roots and thus the same eigenvalues.

How \(A-\lambda I\) is formed

The identity matrix \(I\) has ones on the diagonal and zeros elsewhere. Subtracting \(\lambda I\) from \(A\) affects only the diagonal: \[ A-\lambda I= \begin{bmatrix} a_{11}-\lambda & a_{12} & \cdots \\ a_{21} & a_{22}-\lambda & \cdots \\ \vdots & \vdots & \ddots \end{bmatrix}. \] Off-diagonal entries stay the same; each diagonal entry becomes \(a_{ii}-\lambda\).

Taking the determinant of this matrix produces a polynomial of degree \(n\). The calculator expands that polynomial and shows the coefficients explicitly.

2×2 example

For \[ A=\begin{bmatrix}2&1\\1&2\end{bmatrix}, \] we form \[ A-\lambda I=\begin{bmatrix}2-\lambda & 1\\ 1 & 2-\lambda\end{bmatrix}. \] Then \[ p(\lambda)=\det(A-\lambda I)=(2-\lambda)^2-1=\lambda^2-4\lambda+3. \] Solving \(p(\lambda)=0\) gives \(\lambda=1\) and \(\lambda=3\), so the eigenvalues are \(\{1,3\}\).

Root finding: exact vs numeric

Once you have the polynomial \(p(\lambda)\), eigenvalues are found by solving \(p(\lambda)=0\). For degree 2, a closed-form solution (the quadratic formula) always exists. For degree 3 and 4, closed forms exist in theory, but they are often too complicated to be useful in a learning tool.

This calculator therefore uses a practical approach:

Exact attempt (small cases): quadratic formula for degree 2, and a rational-root factor check for some cubics with integer-like coefficients.
Numeric roots (general): a complex root finder that approximates all roots, including complex conjugate pairs.

A small “root check” \(|p(\lambda_k)|\) indicates the found root is consistent with the polynomial.

Graphing \(p(x)\) on the real axis

The graph plots \(y=p(x)\) for real \(x\). When an eigenvalue is real, it appears where the curve crosses \(y=0\). Complex eigenvalues do not correspond to real-axis crossings; instead, they come in conjugate pairs when \(A\) has real entries.

The calculator automatically chooses a real-axis plotting range using a simple bound derived from row sums (related to Gershgorin-type estimates). You can also switch to a manual range if you want to zoom in near suspected roots.

Stability preview (teaser)

Eigenvalues connect directly to stability in many applications. For the continuous-time linear system \[ x'(t)=Ax(t), \] solutions typically behave like sums of terms \(e^{\lambda t}\). If every eigenvalue satisfies \(\Re(\lambda)<0\), solutions decay. If any eigenvalue has \(\Re(\lambda)>0\), there is a direction of exponential growth.

The calculator’s stability preview counts eigenvalues by the sign of \(\Re(\lambda)\). This is a quick indicator, not a full analysis (especially when eigenvalues lie near the imaginary axis).

University note: Jordan form and sensitivity

For repeated eigenvalues, the matrix may or may not be diagonalizable. In more advanced courses, the Jordan form describes how eigenvalues combine with generalized eigenvectors.

Numerically, eigenvalues can be sensitive when the matrix is ill-conditioned or nearly defective. Small perturbations may shift roots noticeably, especially for higher-degree polynomials. That’s one reason practical eigenvalue computations often use QR/SVD-based methods rather than explicitly forming the characteristic polynomial for large matrices.

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