Matrix Exponential

Let ''X'' be a ''n''×''n'' Real or Complex Matrix . The exponential of ''X'', denoted by ''e''^''X'' or exp(''X''), is the ''n''×''n'' matrix given by the Power Series : : $e^X = \sum_{k=0}^\infty{X^k \over k!}.$ The above series always converges, so the exponential of ''X'' is well-defined. Note that if ''X'' is a 1×1 matrix the matrix exponential of ''X'' corresponds with the ordinary exponential of ''X'' thought of as a number. PROPERTIES Let ''X'' and ''Y'' be ''n''×''n'' complex matrices and let ''a'' and ''b'' be arbitrary complex numbers. We denote the ''n''×''n'' Identity Matrix by ''I'' and the Zero Matrix by 0. The matrix exponential satisfies the following properties: $e^0 = I$ . $e^{aX}e^{bX} = e^{(a+b)X}$ . $e^{X}e^{-X} = I$ . If $Y$ is Invertible then $e^{YXY^{-1}} = Ye^XY^{-1}$ . $\det(e^X) = e^{\mbox{tr}(X)}.$ exp(''X''^T) = (''e''^''X'')^T, where ''X''^T denotes the Transpose of ''X''. It follows that if ''X'' is Symmetric then ''e''^''X'' is also symmetric, and that if ''X'' is Skew-symmetric then ''e''^''X'' is Orthogonal . exp(''X''^---) = (''e''^''X'')^---, where ''X''^--- denotes the Conjugate Transpose of ''X''. It follows that if ''X'' is Hermitian then ''e''^''X'' is also Hermitian, and that if ''X'' is Skew-Hermitian then ''e''^''X'' is Unitary . Linear differential equations One of the reasons for the importance of the matrix exponential is that it can be used to solve systems of linear Ordinary Differential Equations . Indeed, it follows from equation (1) below that the solution of : $rac{d}{dt} y(t) = Ay(t), \quad y(0) = y_0,$ where ''A'' is a matrix, is given by : $y(t) = e^{At} y_0. \,$ The matrix exponential can also be used to solve the inhomogeneous equation : $rac{d}{dt} y(t) = Ay(t) + z(t), \quad y(0) = y_0.$ See the Section On Applications Below for examples. There is no closed-form solution for differential equations of the form : $rac{d}{dt} y(t) = A(t) \, y(t), \quad y(0) = y_0,$ where ''A'' is not constant, but the Magnus Series gives the solution as an infinite sum. The exponential of sums We know that the exponential function satisfies $e^{x+y}=e^xe^y$ for any numbers ''x'' and ''y''. The same goes for commuting matrices: If the matrices ''X'' and ''Y'' commute (meaning that ''XY'' = ''YX''), then : $e^{X+Y} = e^Xe^Y. \,$ However, if they do not commute, then the above equality does not necessarily hold. In that case, we can use the Baker-Campbell-Hausdorff Formula to compute $e^{X+Y}$ . The exponential map Note that the exponential of a matrix is always a Non-singular Matrix . The Inverse of ''e''^''X'' is given by ''e''^−''X''. This is analogous to the fact that the exponential of a complex number is always nonzero. The matrix exponential then gives us a map : $\exp \colon M_n(\mathbb C) o \mbox{GL}(n,\mathbb C)$ from the space of all ''n''×''n'' matrices to the General Linear Group , i.e. the Group of all non-singular matrices. In fact, this map is Surjective which means that ''every'' non-singular matrix can be written as the exponential of some other matrix (for this, it is essential to consider the field C of complex numbers and not '''R'''). The Matrix Logarithm gives an inverse to this map. For any two matrices ''X'' and ''Y'', we have
	Where &nbsp&middot&nbsp Denotes An Arbitrary	"http://wwwinformationdelightinfo/encyclopedia/entry/matrix_norm" class="copylinks">Matrix Norm It follows that the exponential map is Continuous and Lipschitz Continuous on Compact subsets of ''M''<sub>''n''</sub>('''C''')

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Matrix Exponential