More explicit formulas for the matrix exponential (Q1361835)

Using elementary methods, formulas are given for the matrix exponential \(e^{tA}\), where \(A\) is a given complex \(n\times n\) matrix. For example: If \(A\) has distinct eigenvalues \(\lambda_1, \dots, \lambda_r\) and the minimal polynomial of \(A\) has the form \(\prod^r_{j=1} (x-\lambda_j)^{m_j}\), then \[ e^{tA}= \sum^r_{k=1} \left[\sum^{m_k-1}_{s=0} f_{k, s} (t)(A- \lambda_kI)^s \right] \prod^r_{j=1,j \neq k} (A- \lambda_j I)^{m_j}. \] Here the functions \(f_{k,s} (t)\) are determined by solving an explicit triangular system of linear equations. The formulas are specialized to the cases \(n=3\) and \(n=4\) yielding explicit expressions for the matrix exponential in terms of the eigenvalues and the minimal polynomial of \(A\), and in terms of the entries of \(A\). To this end, well-known formulas for solutions of the cubic and quartic equations are used.