Eigenvalue and Eigenvector Calculator
Calculate eigenvalues and eigenvectors of a matrix. Find the characteristic polynomial and solve for λ.
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How Eigenvalues and Eigenvectors Work
For a square matrix A, an eigenvalue λ and its corresponding eigenvector v satisfy:
\[ Av = \lambda v \]
This means that when matrix A is applied to vector v, the result is a scalar multiple of v.
To find eigenvalues:
- Calculate the characteristic polynomial: det(A - λI) = 0
- Solve for λ to find eigenvalues
- For each eigenvalue, solve (A - λI)v = 0 to find eigenvectors
Properties:
- Algebraic multiplicity: number of times an eigenvalue appears in the characteristic polynomial
- Geometric multiplicity: dimension of the eigenspace for each eigenvalue
- Sum of eigenvalues = trace of matrix
- Product of eigenvalues = determinant of matrix