Can eigenvalues be complex numbers?

Yes. When the characteristic polynomial has complex roots, the eigenvalues are complex conjugate pairs. This commonly occurs when the matrix represents a rotation. The calculator displays both real and complex eigenvalues.

Eigenvalue Calculator - Find Eigenvalues & Eigenvectors

Q: What is an eigenvalue?

An eigenvalue lambda of a square matrix A is a scalar such that Av = lambda*v for some non-zero vector v (the eigenvector). In other words, multiplying A by v simply scales v by lambda without changing its direction. Eigenvalues reveal fundamental properties of linear transformations.

Q: How do you find eigenvalues?

Eigenvalues are found by solving the characteristic equation det(A - lambda*I) = 0, where I is the identity matrix. For a 2x2 matrix this gives a quadratic equation, and for a 3x3 matrix a cubic equation. The roots of the characteristic polynomial are the eigenvalues.

Q: What are eigenvectors?

An eigenvector v corresponding to eigenvalue lambda satisfies (A - lambda*I)v = 0. It is found by solving this homogeneous system. The eigenvector represents a direction that is preserved (only scaled) by the linear transformation A.

Q: What is the characteristic polynomial?

The characteristic polynomial of a matrix A is det(A - lambda*I), a polynomial in lambda whose roots are the eigenvalues. For a 2x2 matrix it is lambda^2 - trace*lambda + det = 0. For a 3x3 matrix it is a cubic polynomial.

Eigenvalue Calculator

Find eigenvalues and eigenvectors of 2×2 and 3×3 matrices. Enter the matrix elements and see the characteristic polynomial, eigenvalues, and corresponding eigenvectors instantly.

Find Eigenvalues & Eigenvectors

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Eigenvalues & Eigenvectors

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How to Use the Eigenvalue Calculator

Choose the matrix size (2×2 or 3×3), enter the matrix elements, and the calculator will solve the characteristic polynomial to find all eigenvalues. For each real eigenvalue, the corresponding eigenvector is computed and displayed. Complex eigenvalues are shown in a + bi form.

For example, the identity matrix has eigenvalue 1 with multiplicity equal to the matrix size. A rotation matrix will typically produce complex eigenvalues related to the rotation angle.

Understanding Eigenvalues and Eigenvectors

The eigenvalue equation Av = λv says that the matrix A, when applied to the eigenvector v, produces a vector in the same direction scaled by λ. Finding eigenvalues reduces to solving the characteristic polynomial det(A − λI) = 0. For a 2×2 matrix, this is a quadratic equation with a closed-form solution. For 3×3, it requires solving a cubic.

Properties

The sum of all eigenvalues equals the trace (sum of diagonal elements), and the product of all eigenvalues equals the determinant. A matrix is invertible if and only if none of its eigenvalues are zero. Symmetric matrices always have real eigenvalues.

Applications

Eigenvalues and eigenvectors are central to principal component analysis (PCA) in data science, vibration analysis in mechanical engineering, quantum mechanics (observable operators), Google’s PageRank algorithm, stability analysis of dynamical systems, and image compression (SVD).

Frequently Asked Questions

What is an eigenvalue?

A scalar λ such that Av = λv for some non-zero vector v. It represents a scaling factor for the eigenvector direction.

How do you find eigenvalues?

Solve det(A − λI) = 0. For 2×2 this is a quadratic, for 3×3 a cubic equation.

Can eigenvalues be complex?

Yes, when the characteristic polynomial has complex roots. This occurs commonly with rotation matrices.

What is the characteristic polynomial?

det(A − λI), a polynomial whose roots are the eigenvalues of A.

What are eigenvectors used for?

PCA, vibration analysis, quantum mechanics, PageRank, stability analysis, and image compression.

Disclaimer: This calculator is for informational and educational purposes only. Results are estimates and should not be considered professional expert advice. Consult a qualified professional before making decisions based on these calculations. See our full Disclaimer.