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The Diagonalization Theorem
Definition: Let A be an nXn matrix whose distinct eigenvalues are c_1, c_2, ... , c_k. The following statements are equivalent: (a) A is diagonalizable. (b) The union B of the bases of the eigenspaces of A contains n vectors. (c) The algebraic multiplicity of each eigenvalue equals its geometric multiplicity.
Source: Linear Algebra: A Modern Introduction, 3rd edition by David Poole (note-custom edition titled Matrix Algebra)