Diagonalisírozható

Diagonalisírozható is a concept in linear algebra that describes a square matrix that can be transformed into a diagonal matrix through a similarity transformation. A matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that A = PDP⁻¹. The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

The existence of such a transformation implies that the matrix A behaves similarly to a scalar in many respects. For example, calculating powers of a diagonalizable matrix becomes significantly simpler: Aᵏ = PDᵏP⁻¹. This property is crucial in various applications, including solving systems of linear differential equations, analyzing Markov chains, and understanding the behavior of linear transformations.

A key criterion for a matrix to be diagonalizable is that it must have a complete set of linearly independent eigenvectors. For an n x n matrix, this means there must be n linearly independent eigenvectors. If an eigenvalue has an algebraic multiplicity greater than its geometric multiplicity, the matrix is not diagonalizable. Many matrices encountered in practice, particularly those arising from symmetric or Hermitian forms, are diagonalizable. This property simplifies many computational tasks and theoretical analyses in linear algebra and its applications.