Diagonalizable意思

"Diagonalizable" or "diagonalisable" is a term used in linear algebra, particularly in the context of matrices and linear transformations. A matrix or linear transformation is said to be diagonalizable if it can be transformed into a diagonal matrix by a similarity transformation. In other words, there exists a matrix (or linear transformation) that can be multiplied from the left or right by the original matrix to produce a diagonal matrix.

More formally, a square matrix ( A ) is diagonalizable if there exists a diagonal matrix ( D ) and an invertible matrix ( P ) such that [ A = PDP^{-1}, ] where ( P^{-1} ) is the inverse of matrix ( P ). The diagonal entries of ( D ) are the eigenvalues of ( A ), and the columns of ( P ) are the corresponding eigenvectors of ( A ).

A linear transformation ( T ) on a vector space ( V ) is diagonalizable if there exists a basis of ( V ) such that the matrix representation of ( T ) with respect to this basis is a diagonal matrix.

Diagonalizable matrices and transformations are important because they are relatively easy to work with, and many of the properties of the original matrix or transformation can be deduced from the properties of the diagonal matrix. For example, the eigenvalues of a diagonal matrix are simply the entries on the main diagonal, and the eigenvectors are the columns of the matrix ( P ).