eigenarvon

An eigenarvon is a scalar λ associated with a square matrix A such that there exists a nonzero vector v with Av = λv. The vector v is called an eigenarvonvektor. The eigenarvon equation Av = λv can be rewritten as (A − λI)v = 0, so nontrivial solutions v exist precisely when det(A − λI) = 0. The determinant defines the characteristic polynomial p(λ) = det(A − λI); its roots are the eigenarvon values. For an n×n matrix, there are up to n eigenarvon values, counted with algebraic multiplicity; real matrices may have complex eigenarvon values.

If A is real and not defective, eigenarvon values may be complex in conjugate pairs. When A is symmetric (or Hermitian), all eigenarvon values are real and eigenarvonvektors corresponding to distinct eigenarvon values are orthogonal; the spectral theorem applies. A is diagonalizable if it has n linearly independent eigenarvonvektors, i.e., if the sum of the geometric multiplicities equals n, allowing A = PDP^{-1} with D diagonal.

Computation often relies on solving the characteristic polynomial or using numerical methods such as the QR algorithm. For large or sparse matrices, iterative methods like the power method find dominant eigenarvon values. Applications include solving systems of differential equations, stability analysis, principal component analysis, and PageRank, which uses the eigenarvon of a Markov matrix corresponding to eigenvalue 1.

Example: for A = [[2,1],[0,3]], the eigenarvon values are 2 and 3 with eigenarvonvektors (1,0) and (1,1).