Thesis

Exponentials of Skew-Symmetric Matrices in Terms of Their Eigenvalues

The eigenvalues of an $n\times n$ real nonzero skew-symmetric matrix $S$ are purely imaginary or zero. Let the list of distinct purely imaginary eigenvalues of $S$ be $\pm\theta_1i,\dots ,\pm\theta_pi$ such that each $\theta_j>0$. We algebraically demonstrate that the exponential $e^S$ can be expressed in terms of the powers $I_n,S,\dots,S^{n-1}$, where the coefficients are in terms of the distinct values $\theta_j$, by using the method by Gallier and Xu \cite{Gallier and Xu}. Furthermore, the formulas of $e^S$ (in terms of the $\theta_j$s) depend solely on the number of distinct eigenpairs $\pm\theta_ji$ of $S$ and whether zero is an eigenvalue of $S$, but are independent of their algebraic multiplicities. Only the formulas of the $\theta_j$s (in terms of the entries of $S$) depend on the multiplicities of the $\theta_ji$s in the characteristic polynomial of $S$. This allows us to determine that if $n$ is even, then $e^S$ has $n-1$ different cases, and $\frac{n-1}{2}$ cases if $n$ is odd. In this thesis, we calculate all the closed form formulas of $e^S$ for $2\leq n\leq 9$ because we can obtain the eigenvalues of $S$ in terms of its entries up to the case $n=9$ using the linear, quadratic, cubic, and quartic formulas. Nevertheless, the theory allows us to calculate the closed formula of $e^S$ for any arbitrary $n$ assuming the eigenvalues of $S$ are known. Lastly, we implement the formulas obtained in this thesis on our Matlab function \texttt{skewexpm} and compare the orthogonality errors using our formulas on randomly generated skew-symmetric matrices to those obtained by applying Matlab's \texttt{expm}. It turns out that our formulas give a smaller error than \texttt{expm} for over $97\%$ of the time up to size $n=5$, over $92\%$ of the time up to size $n=7$, and for over $ 60\%$ of the time for sizes $n=8$ and $9$ (see Table 4.1). Finally, if we allow the entries of a skew-symmetric matrix to range from $-10^{15}$ to $10^{15}$, we can rely that our closed formulas will have a far better and acceptable error than \texttt{expm}, as our example with $n=9$ illustrates in Figure 5.9.

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