<p>This paper studies sets of matrices induced by quadratic inequalities. In particular, the center and radius of a smallest ball containing the set, called a <i>Chebyshev center</i> and the <i>Chebyshev radius</i>, are studied. In addition, this work studies the <i>diameter</i> of the set, which is the farthest distance between any two elements of the set. Closed-form solutions are provided for a Chebyshev center, the Chebyshev radius, and the diameter of sets induced by quadratic matrix inequalities (QMIs) with respect to arbitrary unitarily invariant norms. Examples of these norms include the Frobenius norm, spectral norm, nuclear norm, Schatten <i>p</i>-norms, and Ky Fan <i>k</i>-norms. In addition, closed-form solutions are presented for the radius of the largest ball <i>within</i> a QMI-induced set. Finally, the paper discusses applications of the presented results in data-driven modeling and control.</p>