= Density of diagonalizable complex matrices
{title2=$\overline{\{A:A\text{ is diagonalizable}\}}=M_n(\mathbb C)$}
Perturb each diagonal position of a <Jordan normal form> matrix by a different arbitrarily small scalar, avoiding the finitely many collisions between diagonal entries. The resulting upper triangular matrix has distinct <eigenvalues>, hence is <diagonalizable>. Conjugate back to approximate any complex matrix. Continuous identities proved for <diagonalizable matrices> can therefore extend to all matrices.
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