Diagonalizing an operator A by matrix S means finding a new basis for the vector space in which A acts, such that A can be represented as a diagonal matrix with respect to this new basis.
Diagonalization simplifies the calculations involving the operator A, making it easier to analyze and manipulate. It also reveals important properties of the operator, such as its eigenvalues and eigenvectors.
To diagonalize an operator A by matrix S, we need to find the eigenvalues and corresponding eigenvectors of A. The eigenvectors then form the columns of the matrix S, and the eigenvalues form the diagonal elements of the diagonal matrix. The resulting diagonal matrix is the diagonalized form of A.
An operator A can be diagonalized by matrix S if it has a full set of linearly independent eigenvectors. This is only possible if A is a square matrix with distinct eigenvalues.
No, diagonalization is not unique. There can be multiple matrices S that can be used to diagonalize an operator A. Furthermore, the diagonal matrix obtained may also not be unique, as the order of the eigenvalues can be changed without affecting the diagonalization.