The dimension of the variety of normal matrices (Q1569307)

The author studies the problem of what is the dimension of the variety \(N_{n}\) of normal \((n\times n)\) matrices. Recall that normal matrices are defined as matrices \(A\) satisfying the relation \(AA^{*}=A^{*}A.\) The dimension of the variety \(N_{n}\) of normal \((n\times n)\) matrices is shown to be \(n(n+1)/2\) for real matrices and \(n(n+1)\) for complex ones. In both cases, \(N_{n}\) is interpreted as a real algebraic variety. It is also proved that \(N_{n}\) is irreducible in the complex case and reducible in the real one.