Eigenwert

Eigenwert, or eigenvalue, is a scalar associated with a square matrix or linear operator that describes how a corresponding nonzero vector is scaled under the transformation. If A is an n-by-n matrix and v ≠ 0 is a vector such that Av = λv, then λ is an eigenvalue of A and v is an eigenvector associated with λ. The pair (v, λ) indicates that the transformation preserves the direction of v up to a scale.

To find eigenvalues, solve the characteristic equation det(A − λI) = 0, where I is the identity matrix. The polynomial p(λ) = det(A − λI) is the characteristic polynomial, of degree n. Its roots are the eigenvalues (possibly complex). The eigenvectors are the nonzero solutions to (A − λI)v = 0, and for a given λ the set of all such v forms the eigenspace. The algebraic multiplicity of an eigenvalue is its multiplicity as a root of p, while the geometric multiplicity is the dimension of the corresponding eigenspace.

Special cases and properties: If A is real and symmetric, all eigenvalues are real and A can be diagonalized by an orthogonal matrix: A = QΛQ^T, with Q orthogonal and Λ diagonal. In general, a matrix is diagonalizable if there are n linearly independent eigenvectors. If not, it has a Jordan normal form. Real matrices may have complex eigenvalues, which come in conjugate pairs.

Applications and computation: Eigenvalues are used in stability analysis, differential equations, and data analysis (e.g., principal component analysis). Numerically, power iteration finds dominant eigenpairs, while the QR algorithm computes all eigenvalues. For non-square matrices, eigenvalues are not defined; singular values are used instead.

Example: A = [[2,1],[1,2]] has eigenvalues 3 and 1 with eigenvectors (1,1) and (1,−1), respectively.