Eigenvalue Calculator

An eigenvalue is a number that tells you how much a matrix stretches or shrinks a vector along a particular direction. If a matrix A multiplied by a vector v gives you the same vector scaled by a number (Av = 5v), then 5 is an eigenvalue and v is the corresponding eigenvector.

A negative discriminant means the matrix has complex conjugate eigenvalues (involving imaginary numbers). This calculator reports both real and imaginary parts. Complex eigenvalues indicate the transformation involves rotation, which is common in oscillating or rotating systems.

For any square matrix, the product of all eigenvalues equals the determinant, and the sum of all eigenvalues equals the trace. For a 2x2 matrix with eigenvalues 5 and 2, the determinant is 10 and the trace is 7.

Yes. A zero eigenvalue means the matrix is singular (not invertible) and its determinant is zero. The matrix collapses at least one dimension of space to zero, meaning the system of equations Ax = b may have no solution or infinitely many solutions.

Eigenvalues are scalar numbers (how much), while eigenvectors are direction vectors (which direction). Together they describe the complete behavior of a linear transformation: each eigenvector points in a direction that only gets scaled (by its eigenvalue) when the matrix is applied.

Eigenvalues are central to PCA (Principal Component Analysis), which finds the most important directions in high-dimensional data. The largest eigenvalues correspond to the directions with the most variance, allowing you to reduce dimensions while keeping the most information.

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