Numerical radius and zero pattern of matrices
نویسنده
چکیده
Let A be an n n complex matrix and r be the maximum size of its principal submatrices with no o¤-diagonal zero entries. Suppose A has zero main diagonal and x is a unit n-vector. Then, letting kAk be the Frobenius norm of A; we show that jhAx;xij (1 1=2r 1=2n) kAk : This inequality is tight within an additive term O n 2 : If the matrix A is Hermitian, then jhAx;xij (1 1=r) kAk : This inequality is sharp; moreover, it implies the Turán theorem for graphs. AMS classi cation: 15A42, 05C50
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