Eigenvalues Eigenvectors and Diagonalization Simple Notes

An eigenvector of a square matrix A is a nonzero vector v that the matrix only stretches, never rotates: A v = lambda v. The stretching factor lambda is called the eigenvalue. Finding these special directions reveals the hidden skeleton of the matrix.

Eigenvalues and eigenvectors matter because they expose a system's natural modes. The vibration frequencies of a bridge, the stable states of a quantum system, the principal directions of a dataset, and even Google's original page ranking are all eigenvalue problems in disguise.

Key Concepts

• Eigenvalue equation: A v = lambda v, with v nonzero; lambda is the eigenvalue and v an eigenvector.
• Characteristic equation: det(A minus lambda I) = 0, a polynomial whose roots are the eigenvalues.
• Eigenspace: all eigenvectors for one eigenvalue plus the zero vector; any nonzero multiple of an eigenvector is again an eigenvector.
• Diagonalization: if A has enough independent eigenvectors, A = P D P inverse, where D is diagonal with the eigenvalues and P's columns are the eigenvectors.
• Trace and determinant: the trace of A equals the sum of eigenvalues, and the determinant equals their product.

Formulas

• Characteristic polynomial: det(A minus lambda I) = 0.
• For a 2-by-2 matrix [[a, b], [c, d]]: lambda squared minus (trace) lambda + (determinant) = 0.
• Diagonalization: A = P D P inverse, so A to the power n = P D to the power n P inverse, making matrix powers trivial.
• Eigenvalues of a triangular matrix are simply its diagonal entries.
• Symmetric matrices have real eigenvalues and orthogonal eigenvectors.

Worked Example

Find the eigenvalues and eigenvectors of A = [[4, 1], [2, 3]].

1. Form A minus lambda I = [[4 minus lambda, 1], [2, 3 minus lambda]] and take the determinant: (4 minus lambda)(3 minus lambda) minus 2 = lambda squared minus 7 lambda + 10.
2. Factor: lambda squared minus 7 lambda + 10 = (lambda minus 5)(lambda minus 2), so the eigenvalues are lambda1 = 5 and lambda2 = 2.
3. For lambda = 5: solve [[minus 1, 1], [2, minus 2]] v = 0, giving minus x + y = 0, so v1 = (1, 1).
4. For lambda = 2: solve [[2, 1], [2, 1]] v = 0, giving 2x + y = 0, so v2 = (1, minus 2).
5. Check: A times (1,1) = (5, 5) = 5 times (1,1), and A times (1, minus 2) = (2, minus 4) = 2 times (1, minus 2). Both correct.

Common Misconceptions

• "Each eigenvalue has exactly one eigenvector." Any nonzero scalar multiple counts, so each eigenvalue owns a whole line (or plane) of eigenvectors.
• "Every matrix can be diagonalized." Defective matrices, like [[0, 1], [0, 0]], lack enough eigenvectors and cannot be diagonalized.
• "Eigenvalues are always real." Only for symmetric (Hermitian) matrices; a rotation matrix has complex eigenvalues, which encode the rotation angle.

Real-World Applications

• Coupled oscillators: normal modes of molecules and bridges are eigenvectors of the system's matrix.
• Quantum mechanics: measurable energies are eigenvalues of the Hamiltonian operator.
• Data science: principal component analysis (PCA) keeps the eigenvectors with the largest eigenvalues to compress data.

Eigenvalues and eigenvectors distill a matrix down to the directions it treats simply and the factors by which it stretches them. Diagonalization then turns hard matrix problems into easy scalar ones.