Eigenvalues and Eigenvectors 3×3 Matrix Calculator
This calculator finds the eigenvalues and eigenvectors of a 3×3 matrix. It’s ideal for students, engineers, and scientists working with linear algebra, physics, and advanced mathematics where matrix diagonalization and spectral analysis are required.
3×3 Matrix Eigenvalues and Eigenvectors Solver
Eigenvalues and Eigenvectors Formula for 3×3 Matrix
Where:
- $$A$$ is the 3×3 matrix
- $$\lambda$$ represents an eigenvalue
- $$I$$ is the identity matrix
- $$\vec{v}$$ is the corresponding eigenvector
The characteristic equation $$\det(A – \lambda I) = 0$$ is solved for $$\lambda$$, and the resulting $$\lambda$$ values are used to compute the eigenvectors by solving the homogeneous system.
Eigenvalues and Eigenvectors 3×3 – Calculation Example
$$A = \begin{bmatrix} 6 & 2 & 1 \\ 2 & 3 & 1 \\ 1 & 1 & 1 \end{bmatrix} $$
Solve:
$$\det(A – \lambda I) = 0 $$
Characteristic polynomial:
$$\lambda^3 – 10 \lambda^2 + 27 \lambda – 18 = 0 $$
Solutions (eigenvalues):
$$\lambda_1 = 1, \quad \lambda_2 = 3, \quad \lambda_3 = 6$$
Find eigenvectors by solving:
$$(A – \lambda I)\vec{v} = 0$$
for each \(\lambda\).
Eigenvalues and eigenvectors are key in linear transformations, stability analysis, vibration modes, and quantum mechanics. Our calculator simplifies the process of solving the characteristic equation and finding eigenvectors for any 3×3 matrix.