To know whether the eigenvectors meet this condition, it is enough that the determinant of the matrix P is nonzero, which means that the matrix has maximum rank. That is because matrix P is formed by the eigenvectors of that matrix. A square matrix of order n is diagonalizable if it has n linearly independent eigenvectors, in other words, if these vectors form a basis.There are three ways to know whether a matrix is diagonalizable: Not all matrices are diagonalizable, only matrices that meet certain characteristics can be diagonalized. And, logically, P is an invertible matrix. Thus, matrix A and matrix D are similar matrices. Matrix P acts as a change of basis matrix, so in reality with this formula we are actually changing basis to matrix A so that the matrix becomes a diagonal matrix (D) in the new basis. Where A is the matrix to be diagonalized, P is the matrix whose columns are the eigenvectors of A, P -1 its inverse matrix, and D is the diagonal matrix composed by the eigenvalues of A. The mathematical relation between a matrix and its diagonalized matrix is: A diagonalizable matrix is a square matrix that can be transformed into a diagonal matrix, that is, a matrix filled with zeros except for the main diagonal.
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