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The vibration of structures is governed by a set of second-order ordinary differential equations in which the ( f ) coefficient matrices are real. These equations often produce both complex roots and complex modes. In the established method for computing the roots, the eigenvalue problem is solved for a certain (2 f 2 ) matrix whose form is such that it contains an ( f ) submatrix of zeros as one of the two diagonal ( f ) blocks. This very special form has substantial significance in the modes and roots that emerge. For systems having no real roots, a part of this significance has already been identified by the authors in the form of a relationship between the real and imaginary parts of complex modes. This article extends this significance to the point where the equation normally used in computing the complete set of characteristic roots and vectors is transformed to another very compact form. One of the attractions of this new form is that all of the numbers involved are real—although some or all of the roots and vectors may be complex. The new form has several potential applications, including providing new methods for examining the sensitivity of solutions to perturbations, achieving realizations of second-order systems from partial knowledge of the roots and modes, and forming the basis for a new solution method for obtaining the characteristic roots and vectors of self-adjoint second-order systems.