Eigenvalues and Eigenvectors Examples
These 2×2 examples show how the characteristic polynomial leads to eigenvalues—and why repeated or complex roots need extra care.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 20, 2026 · Editorial method
Example 1: symmetric matrix
For A = [[2,1],[1,2]], det(A−λI)=(2−λ)²−1=λ²−4λ+3. The eigenvalues are 3 and 1. Corresponding directions are ⟨1,1⟩ and ⟨1,−1⟩, which are orthogonal.
Example 2: triangular matrix
For A = [[4,2],[0,1]], the eigenvalues are visible on the diagonal: 4 and 1. Solving (A−4I)v=0 gives direction ⟨1,0⟩; solving (A−I)v=0 gives a direction such as ⟨−2,3⟩.
Example 3: repeated but diagonalizable
For A = [[5,0],[0,5]], λ=5 repeats. Since A−5I is the zero matrix, every nonzero vector is an eigenvector. Two independent directions are available.
Example 4: repeated and defective
For A = [[5,1],[0,5]], λ=5 also repeats, but (A−5I)v=0 forces the second component to zero. Only one independent eigendirection exists, so this matrix is not diagonalizable.
Example 5: complex pair
For A = [[0,−1],[1,0]], the characteristic equation is λ²+1=0, giving λ=±i. This rotation matrix has no real eigenvector direction.
Check your own matrix
Use the 2×2 eigenvalue and eigenvector calculator for trace, determinant, roots, and representative real eigenvectors. Then verify each pair with Av=λv.