Eigenvalue Calculator

Use this free eigenvalue and eigenvector calculator to find eigenvalues, eigenvectors, and the characteristic polynomial of any square matrix. Get step-by-step solutions with detailed explanations. Essential for linear algebra and differential equations.

What are Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors reveal fundamental properties of linear transformations. When a matrix \( A \) acts on an eigenvector \( \mathbf{v} \), the result is simply a scaled version of \( \mathbf{v} \). The scaling factor is the eigenvalue \( \lambda \).

$$ A\mathbf{v} = \lambda\mathbf{v}$$

This equation says that the eigenvector \( \mathbf{v} \) only changes in magnitude (not direction) when transformed by \( A \).

The Characteristic Polynomial

To find eigenvalues, solve the characteristic equation:

$$\det(A – \lambda I) = 0$$

This determinant produces a polynomial in \( \lambda \). The roots of this polynomial are the eigenvalues.

For a 2×2 matrix, the characteristic polynomial is quadratic. For a 3×3 matrix, it’s cubic.

Finding Eigenvectors

Once you have an eigenvalue \( \lambda \), find the eigenvector by solving:

$$(A – \lambda I)\mathbf{v} = \mathbf{0}$$

This is a homogeneous system. The solution space (null space of \( A – \lambda I \)) gives the eigenvectors corresponding to \( \lambda \).

Properties

Trace and Determinant

• Sum of eigenvalues = \( \text{trace}(A) \) (sum of diagonal elements)
• Product of eigenvalues = \( \det(A) \)

Algebraic vs Geometric Multiplicity

• Algebraic multiplicity: how many times \( \lambda \) appears as a root
• Geometric multiplicity: dimension of the eigenspace

Real vs Complex Eigenvalues

Real matrices can have complex eigenvalues, which always come in conjugate pairs.

Applications

Principal Component Analysis (PCA)

Eigenvectors of the covariance matrix identify directions of maximum variance in data.

Stability Analysis

Eigenvalues determine whether a system is stable:

• All \( |\lambda| < 1 \): stable (discrete systems)
• All \( \text{Re}(\lambda) < 0 \): stable (continuous systems)

Quantum Mechanics

Observable quantities correspond to eigenvalues of Hermitian operators.

Google’s PageRank

The importance of webpages is determined by the dominant eigenvector of the link matrix.

Vibration Analysis

Natural frequencies of mechanical systems are eigenvalues of the system matrix.

Special Cases

Matrix Type	Eigenvalue Property
Symmetric	All real eigenvalues
Orthogonal	\(
Positive definite	All \( \lambda > 0 \)
Nilpotent	All \( \lambda = 0 \)
Idempotent	\( \lambda = 0 \) or \( 1 \)

Diagonalization

A matrix \( A \) is diagonalizable if it has \( n \) linearly independent eigenvectors. Then:

$$A = PDP^{-1}$$

Where \( D \) is diagonal (eigenvalues on diagonal) and \( P \) contains eigenvectors as columns.

An eigenvector keeps its direction when a matrix transformation is applied. The matrix may stretch it, shrink it, reverse it, or collapse it to zero. The corresponding scale factor is the eigenvalue.

How to use this calculator

Enter a square matrix. Eigenvalues are defined for square linear transformations. The calculator forms the characteristic equation \(\det(A-\lambda I)=0\), solves for \(\lambda\), then finds the null space of \(A-\lambda I\) for each eigenvector.

Keep exact entries when possible. Decimal rounding can split a repeated eigenvalue or make a nearly dependent eigenvector calculation look inconsistent.

1. Subtract \(\lambda\) from the diagonal entries.
2. Compute the characteristic determinant.
3. Solve the characteristic polynomial for eigenvalues.
4. For each eigenvalue, solve \((A-\lambda I)v=0\) and choose a nonzero vector.

Worked example

For \(A=\begin{pmatrix}2&1\\1&2\end{pmatrix}\), the characteristic polynomial is \((2-\lambda)^2-1\), giving eigenvalues 3 and 1.

For \(\lambda=3\), an eigenvector is (1,1). For \(\lambda=1\), an eigenvector is (1,-1). Multiplying confirms \(A(1,1)=3(1,1)\) and \(A(1,-1)=1(1,-1)\).

How to read the result

The trace equals the sum of eigenvalues and the determinant equals their product, counting multiplicity. These are strong checks for a computed result.

A repeated eigenvalue may have enough independent eigenvectors for diagonalization, or it may not. Algebraic multiplicity comes from the characteristic polynomial; geometric multiplicity comes from the eigenspace dimension.

Common mistakes to avoid

• Using \(A-\lambda\) instead of \(A-\lambda I\).
• Accepting the zero vector as an eigenvector.
• Assuming every real matrix has only real eigenvalues.
• Assuming a repeated eigenvalue automatically supplies several independent eigenvectors.

How to verify the result

For every reported pair, multiply the matrix by the eigenvector and compare it with the eigenvalue times that vector. The entries should agree apart from rounding. The zero vector is never an eigenvector.

Use the trace and determinant as global checks. For a square matrix, the eigenvalues add to the trace and multiply to the determinant when multiplicity is counted. A mismatch usually points to a copied matrix entry or a missed complex root.

Keep exact fractions and radicals as long as possible. Decimal rounding is especially risky with repeated or nearly repeated eigenvalues because a small perturbation can make one value appear as several nearby values.

Limits of the calculation

Large numerical matrices need stable algorithms such as QR iteration rather than symbolic determinant expansion. Ill-conditioned problems can move eigenvalues noticeably under tiny input changes.

For a non-square matrix, singular values and singular vectors often play the role you actually need.

Related calculators

Use Matrix Operations Calculator, Row Reduction Calculator, System of Linear Equations Calculator when the next part of the problem needs a different method.

Useful linear algebra books and tools

For eigenvalues, connect the characteristic polynomial to invariant directions instead of treating the roots as isolated numbers. Matrix output becomes much easier to understand when you connect the row operations to vectors, transformations, and systems of equations.

• Learn the structure: Browse linear algebra textbooks and workbooks on Amazon. Prioritize books that explain geometric meaning alongside computational methods.
• Check numerical work: Browse scientific calculators on Amazon. A calculator can catch arithmetic slips, though it should not replace a written row-reduction trail.

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Frequently Asked Questions