Eigenvalues and Eigenvectors 2×2 Matrix Calculator
This calculator computes the eigenvalues and eigenvectors of a 2×2 matrix. It is perfect for students, engineers, and scientists working with linear algebra, physics, and systems analysis where eigen decomposition is required.
2×2 Matrix Eigenvalues and Eigenvectors Solver
Eigenvalues and Eigenvectors Formula for 2×2 Matrix
Where:
- $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ is the 2×2 matrix
- $$\lambda$$ are the eigenvalues
- $$\vec{v}$$ are the corresponding eigenvectors
The characteristic equation is solved for $$\lambda$$, then eigenvectors are found by solving the system $$(A – \lambda I)\vec{v} = 0$$.
Eigenvalues and Eigenvectors 2×2 – Calculation Example
$$A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix} $$
$$\det\begin{bmatrix} 4 – \lambda & 2 \\ 1 & 3 – \lambda \end{bmatrix} = (4 – \lambda)(3 – \lambda) – 2 \cdot 1 = \lambda^2 – 7 \lambda + 10 = 0 $$
Solutions (eigenvalues):
$$\lambda_1 = 5, \quad \lambda_2 = 2$$
Find eigenvectors by solving:
$$(A – \lambda I)\vec{v} = 0$$
for each $$\lambda$$.
Eigenvalues and eigenvectors of a 2×2 matrix are essential in solving systems of equations, stability analysis, and understanding linear transformations. Our calculator helps find them quickly and accurately, avoiding manual calculation errors.