Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6416448 | Linear Algebra and its Applications | 2014 | 14 Pages |
For an n-by-n complex matrix A, we define its zero-dilation index d(A) as the largest size of a zero matrix which can be dilated to A. This is the same as the maximum k (⩾1) for which 0 is in the rank-k numerical range of A. Using a result of Li and Sze, we show that if d(A)>â2n/3â, then, under unitary similarity, A has the zero matrix of size 3d(A)â2n as a direct summand. It complements the known fact that if d(A)>ân/2â, then 0 is an eigenvalue of A. We then use it to give a complete characterization of n-by-n matrices A with d(A)=nâ1, namely, A satisfies this condition if and only if it is unitarily similar to Bâ0nâ3, where B is a 3-by-3 matrix whose numerical range W(B) is an elliptic disc and whose eigenvalue other than the two foci of âW(B) is 0. We also determine the value of d(A) for any normal matrix A and any weighted permutation matrix A with zero diagonals.