Eigenvalues and Eigenvectors: Explained with Examples

An eigenvector of a square matrix A is a nonzero vector A only stretches, never rotates: Av = lambda v for some scalar lambda, the eigenvalue. Most vectors get twisted into new directions by A, but eigenvectors stay on their own line, merely scaled, making them the natural axes of the transformation.

They reveal hidden structure: the directions where a system grows or shrinks. Engineers find bridge resonances this way, Google once ranked pages with them, and data scientists compress images with them. In each case the important action happens along eigenvectors.

Key Concepts

• Eigenvector: nonzero v with Av = lambda v. Zero is excluded since it works trivially for every lambda.
• Eigenvalue: the stretch factor lambda; negatives flip direction, values between minus 1 and 1 shrink.
• Eigenspace: all eigenvectors for one lambda plus zero, equal to ker(A minus lambda I).
• Characteristic polynomial: det(A minus lambda I), degree n, roots are the eigenvalues.
• Multiplicities: algebraic (root multiplicity) vs geometric (eigenspace dimension); geometric is between 1 and algebraic.

Formulas

• Eigenvalue equation: det(A minus lambda I) = 0; then solve (A minus lambda I)v = 0 per eigenvalue.
• Trace: sum of eigenvalues = sum of diagonal entries.
• Determinant: product of eigenvalues = det(A); A invertible exactly when 0 is not an eigenvalue.
• Powers: Av = lambda v implies A to the kth v = lambda to the kth v.

Worked Example

For A = [[4,1],[2,3]]: det(A minus lambda I) = (4 minus lambda)(3 minus lambda) minus 2 = lambda squared minus 7 lambda + 10 = (lambda minus 5)(lambda minus 2), so eigenvalues 5 and 2. Check: trace 5+2 = 7 = 4+3; det 10 = 12 minus 2. For lambda = 5, [[minus 1,1],[2,minus 2]](x,y) = 0 gives y = x, so v = (1,1); verify A(1,1) = (5,5) = 5(1,1). For lambda = 2, [[2,1],[2,1]](x,y) = 0 gives y = minus 2x, so v = (1,minus 2); verify A(1,minus 2) = (2,minus 4) = 2(1,minus 2).

Common Misconceptions

• Eigenvalues must be real. Rotations have complex eigenvalues; the polynomial always has n roots over C, possibly none over R.
• Every matrix has n independent eigenvectors. [[2,1],[0,2]] has eigenvalue 2 twice but a 1D eigenspace: defective, not diagonalisable.
• Eigenvectors are unique. Any nonzero multiple works; the eigenspace, not the vector, is canonical.

Real-World Applications

• Engineering: eigenvalues of stiffness matrices give a structure's natural vibration frequencies.
• PageRank: Google's original ranking used the dominant eigenvector of the web link matrix.
• Compression: PCA keeps eigenvectors of largest eigenvalue, the directions of greatest variation.

Eigenvectors are the directions a matrix scales without rotating; eigenvalues are the scale factors. Solve det(A minus lambda I) = 0, then each homogeneous system, verify by direct multiplication, and never accept the zero vector as an eigenvector.