Diagonalization is a process in linear algebra where a square matrix is transformed into a diagonal matrix by finding a basis of eigenvectors for the matrix.
Diagonalization is important because it simplifies calculations involving linear transformations and makes it easier to understand the behavior of the matrix.
The eigenvalues of a matrix can be found by solving the characteristic equation det(A-λI)=0, where A is the matrix and λ is the eigenvalue. The corresponding eigenvectors can then be found by plugging in the eigenvalues into the equation (A-λI)v=0 and solving for v.
No, not all matrices can be diagonalized. A matrix can only be diagonalized if it has n linearly independent eigenvectors, where n is the size of the matrix.
The diagonal elements of a diagonalized matrix are the eigenvalues of the original matrix. They represent the scaling factors of the corresponding eigenvectors in the transformation represented by the matrix.