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ad sinh (A) 2 A sinh (A) b-A cosh(A) ---cosh (A) + -sinh ( A ) A ad sinh (A) C -tional kinematics, see [7]. An extensive treatment of this topic including the relationship between rotations and quaternions is discussed in [ l ] .. THE EXPONENTIAL OF Abstmct-The matrix exponential plays a central role in linear sys-In this section, we derive formulas for the exponential of a either the eigenvalues of A or the entries of A . The results are specialized to the case in which A is a real matrix. Let R and C denote the real and complex numbers respectively. temS a d Control theory. In this note, we give explicit formulas for 2 x 2 complex matrix A . Formulas are given in terms of computing the exponential of some special matrices.

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An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

The energy of a graph is the sum of the singular values of its adjacency matrix. We are interested in how the energy of a graph changes when edges are deleted. Examples show that all cases are possible: increased, decreased, unchanged. Our goal is to find possible graph theoretical descriptions and to provide an infinite family of graphs for each case. The main tool is a singular value inequality for complementary submatrices and its equality case.