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Theorem

An n×n n \times n matrix A A is diagonalizable if and only if A A has n n linearly independent eigenvectors.

In fact, A=PDP1 A = PDP^{-1} , with D D a diagonal matrix, if and only if the columns of P P are n n linearly independent eigenvectors of A A . In this case, the diagonal entries of D D are eigenvalues of A A that correspond, respectively, to the eigenvectors in P P .

Theorem

An n×n n \times n matrix with n n distinct eigenvalues is diagonalizable.

Theorem

Let A A be an n×n n \times n matrix whose distinct eigenvalues are λ1,,λp \lambda_1, \dots, \lambda_p .

a. For 1kp 1 \leq k \leq p , the dimension of the eigenspace for λk \lambda_k is less than or equal to the multiplicity of the eigenvalue λk \lambda_k .
b. The matrix A A is diagonalizable if and only if the sum of the dimensions of the distinct eigenspaces equals n n , and this happens if and only if the dimension of the eigenspace for each λk \lambda_k equals the multiplicity of λk \lambda_k .
c. If A A is diagonalizable and Bk \mathcal{B}_k is a basis for the eigenspace corresponding to λk \lambda_k for each k k , then the total collection of vectors in the sets B1,,Bp \mathcal{B}_1, \dots, \mathcal{B}_p forms an eigenvector basis for Rn \mathbb{R}^n .