Eigenvalues & Eigenvectors

Input Matrix A

Calculation Results & Process

💡 Geometric Meaning of Eigenvalues & Eigenvectors

• Eigenvalues & Eigenvectors: Under a linear transformation represented by a matrix $A$, eigenvectors $\vec{v}$ are special non-zero vectors whose direction remains unchanged, only scaled by a factor of $\lambda$ (the eigenvalue): $$A \vec{v} = \lambda \vec{v} \quad (\vec{v} \neq \vec{0})$$
• Characteristic Equation: For $(A - \lambda I)\vec{v} = \vec{0}$ to have non-zero solutions for $\vec{v}$, the matrix $A - \lambda I$ must not be invertible. Therefore, its determinant must equal 0: $$\det(A - \lambda I) = 0$$ Solving this equation yields the eigenvalues $\lambda$.
• Eigenvector Calculation: Substitute each calculated eigenvalue $\lambda_i$ back into the homogeneous equation $(A - \lambda_i I)\vec{v} = \vec{0}$, and solve the system using row reduction to find the basis of eigenvectors.