Eigenvalue assignment for real matrices of specified form

The purpose of this note is to report on various existence conditions found in connection with the problem of assigning the eigenvalues of a real
n
th-order matrix constrained by the specification of certain zero entries. This problem finds applications in many areas, but it was originally motivated by attempts to obtain desired responses from an
n
th-order dynamic system governed by a set of linear homogeneous differential equations of the form
\dot{x} = Ax
where
A
is the constant real system matrix to be found.