1.1.13 Matrix Powers and Logarithms

The ex­po­nen­tial of a matrix $M$, is de­fined as the fol­low­ing in­fi­nite series

which con­verges to the re­sult­ing matrix $N$. We can define the matrix M, as the matrix log­a­rithm of N, since $e^{M} = N.$In terms of the ex­is­tence of the matrix log­a­rithm, com­plex ma­tri­ces are straight­for­ward, where the log­a­rithm does exist if the matrix is in­vert­ible. Log­a­rithms may not nec­es­sar­i­ly be unique, but if there are no eigen­val­ues $\in \mathbb{R}^{-}$, a unique matrix log­a­rithm, termed the prin­ci­pal log­a­rithm, lies in region $z\in \mathbb{C}| -\pi < z < \pi|$. For real ma­tri­ces, a matrix log­a­rithm exists if that matrix is in­vert­ible, and for all neg­a­tive eigen­val­ues $- \lambda$, the Jordan blocks are in even num­bers.

In Lie theory, we have the ex­po­nen­tial map­ping from the Lie al­ge­bra $\mathfrak{g}$, to its Lie group $e^{\mathfrak{g}} \rightarrow G$, and the cor­re­spond­ing log­a­rithm map­ping from the Lie group $G$, to the Lie al­ge­bra${\ ln}(G)\rightarrow \mathfrak{g}.$In this book, the matrix groups that we con­sid­er are di­ag­o­nal­iz­able, and we can take ad­van­tage of this to com­pute matrix ex­po­nen­tials and matrix log­a­rithms. We can take an ini­tial di­ag­o­nal­iz­able Lie al­ge­bra matrix $\mathfrak{g}$, and ex­po­nen­ti­ate it

for $V$, a matrix formed by eigen­vec­tor columns of $\mathfrak{g}$, and $\mathfrak{d}$, the di­ag­o­nal­ized matrix of its eigen­val­ues. We can now com­pute both the ex­po­nen­tial and the re­sult­ing log­a­rithm, as

and

With the matrix ex­po­nen­tial and the matrix log­a­rithm in mind, we now have the tools to con­sid­er matrix powers. For a com­plex matrix M, raised to a com­plex power, we can com­pute com­plex powers of $M$, as

Taking the di­ag­o­nal­iz­able matrix $M$

we find that

for $adj(V)$, rep­re­sent­ing the ad­ju­gate of the $V$-matrix, where $Vadj(V)$ can be seen to equal the prod­uct $\det(V)I$. We then have

for $D$, the di­ag­o­nal matrix of eigen­val­ues from $M$, with which, we can find $\ln(M)$

With the result for $\ln(M)$, we have what is re­quired to cal­cu­late the matrix $M$, raised to a com­plex power $M^{a + ib}$, by the fol­low­ing

find­ing

A more straight­for­ward ex­am­ple would be to cal­cu­late

The math­e­mat­ics in­volved in rais­ing a number or matrix to imag­i­nary/com­plex powers is an im­por­tant con­cept to know. We use it when con­sid­er­ing the angle spaces later in the book, which are not bound by the con­straints of geo­met­ric spaces as they are nor­mal­ly treat­ed.