When a matrix is diagonalisable?
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A (square) matrix is diagonalizable if and only if it has a full set of eigenvectors. That is, there exist a basis for the vector space consisting of eigenvectors of the matrix. Since distinct eigenvalues correspond to independent eigenvectors, an n by n matrix with n distinct eigenvalues is diagonalizable but there a matrix with duplicate eigenvalues may still be diagonalizable if there exist enough independent eigenvectors.