Eigenvalue and Eigenvector Calculator

Math Linear Algebra • Linear Transformations and Eigenvalues

Written by STEM Calculators Team

Compute eigenvalues \(\lambda\) from \(\det(A-\lambda I)=0\), and eigenvectors \(v\neq 0\) from \((A-\lambda I)v=0\). Supports 2×2 (exact quadratic) and 3×3 (cubic/Cardano over \(\mathbb{C}\)) with multiplicities, normalized eigenvectors, and a spectrum plot.

Matrix size

Tolerance

Display decimals

Steps

Normalize eigenvectors (\(\|v\|=1\)) Scale so first nonzero component is 1 Show residual checks \(\|(A-\lambda I)v\|\) Show spectrum plot (complex plane) Show stretching directions visual (2D real eigs)

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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Trace / determinant

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Eigenvalues (with algebraic multiplicity)

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Spectral radius \(\rho(A)=\max|\lambda|\)

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Eigenvectors (basis of each eigenspace)

Geometric multiplicity = dimension of \(\mathrm{Null}(A-\lambda I)\).

Ready

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

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Theory

Eigenvalues and eigenvectors

For a square matrix \(A\in\mathbb{R}^{n\times n}\), an eigenvalue \(\lambda\) and a nonzero eigenvector \(v\neq 0\) satisfy \[ Av=\lambda v. \] This means the transformation \(A\) preserves the direction \(v\) and scales it by \(\lambda\).

Characteristic polynomial: \(\det(A-\lambda I)=0\)

Rewrite \(Av=\lambda v\) as \((A-\lambda I)v=0\). A nontrivial solution exists exactly when \(A-\lambda I\) is singular: \[ \det(A-\lambda I)=0. \] The left-hand side is the characteristic polynomial \(p(\lambda)\). Its roots are the eigenvalues (over \(\mathbb{C}\) if needed).

For \(2\times2\), \[ p(\lambda)=\lambda^2-\mathrm{tr}(A)\lambda+\det(A). \] For \(3\times3\), \[ p(\lambda)=\lambda^3-(\mathrm{tr}A)\lambda^2+c_1\lambda-\det(A), \] where \[ c_1=\frac{(\mathrm{tr}A)^2-\mathrm{tr}(A^2)}{2}. \]

Finding eigenvectors: null space of \(A-\lambda I\)

For each eigenvalue \(\lambda\), eigenvectors are the nonzero solutions of \[ (A-\lambda I)v=0. \] This solution set is the eigenspace \[ E_\lambda=\mathrm{Null}(A-\lambda I). \] Any basis of \(E_\lambda\) is a valid set of eigenvectors for \(\lambda\).

Algebraic vs geometric multiplicity

The number of times \(\lambda\) appears as a root of \(p(\lambda)\) is its algebraic multiplicity. The dimension \(\dim(E_\lambda)\) is its geometric multiplicity. Always, \[ 1\le \dim(E_\lambda)\le \text{(algebraic multiplicity)}. \] If the geometric multiplicities add up to \(n\), the matrix is diagonalizable (over \(\mathbb{C}\)).

Complex eigenvalues

Real matrices can have complex eigenvalues (e.g., rotations). In that case eigenvectors generally live in \(\mathbb{C}^n\). Plotting eigenvalues in the complex plane helps visualize the spectrum.

Stretching directions and repeated application

In 2D with real eigenvalues and distinct eigendirections, applying \(A\) repeatedly tends to amplify the component of a vector along the eigendirection corresponding to the largest \(|\lambda|\). This is the intuition behind “power iteration”: \[ x_{k+1}=Ax_k \quad (\text{often normalized each step}). \] The stretching-visual shows eigenvector lines and how many starting directions evolve under repeated application.

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