diagonalisálható

In linear algebra, a square matrix is called diagonalizable if it is similar to a diagonal matrix. This means that there exists an invertible matrix P such that P⁻¹AP is a diagonal matrix D. The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

A matrix is diagonalizable if and only if it has a full set of linearly independent eigenvectors. For an n x n matrix, this means that there are n linearly independent eigenvectors. This condition is met if the matrix has n distinct eigenvalues. However, even if eigenvalues are repeated, the matrix can still be diagonalizable, provided that the geometric multiplicity of each eigenvalue (the dimension of its eigenspace) equals its algebraic multiplicity (the number of times it appears as a root of the characteristic polynomial).

Diagonalizable matrices have several important properties. For instance, powers of a diagonalizable matrix A can be easily computed: if A = PDP⁻¹, then Aᵏ = PDᵏP⁻¹. The powers of a diagonal matrix D are simply the powers of its diagonal entries. This simplifies many calculations in areas like differential equations and discrete dynamical systems.

The concept of diagonalizability is crucial for understanding the behavior of linear transformations and for simplifying complex matrix operations. It allows for a transformation to be represented in a simpler, coordinate-independent form.