Derivatives of the characteristic polynomial, trace, and determinant with respect to a matrix
John Brewer · IEEE Transactions on Automatic Control · 1979
Matrix calculus [2], [7] is used to derive formulas for the derivatives of the coefficients of the characteristic polynomial with respect to any matrix of physical parameters. Two of these derivatives may be related to the matrix derivatives of the determinant and the negative trace. The use of these derivative formulas is restricted to "nonderogatory" matrices. This is a large class of matrices which includes all matrices with nonrepeated eigenvalues. The use and validity of the formulas is demonstrated by showing that they provide a correct and well-known result for a particular special case. Higher order matrix derivatives are easily obtained once formulas for the first-order derivatives are known.