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Question Number 2546 by Rasheed Soomro last updated on 22/Nov/15

$$\mathbf{If}\mathrm{A}\mathrm{and}\mathrm{B}\mathrm{are}\mathrm{two}\mathrm{matrices}\mathrm{of}\mathrm{suitable}\mathrm{order}$$$$\mathrm{does}\mathrm{there}\mathrm{exist}\mathrm{definition}\mathrm{for}{\mathrm{A}}^{\mathrm{B}}?$$

Commented by Yozzi last updated on 22/Nov/15

$$A^B=e^{\left(lnA\right)B}ifAandBaresqaure$$$$andnon-singular.$$$$e^{\left(lnA\right)B}=1+\left(lnA\right)B+\frac{{\left(\left(lnA\right)B\right)}^2}{2!}+\frac{{\left(\left(lnA\right)B\right)}^3}{3!}+...$$$$\therefore A^B=\underset{r=0}{\sum ^{\mathrm{\infty }}}\frac{1}{r!}{\left[\left(lnA\right)B\right]}^r\left(Matrixexponential\right)$$

Commented by Rasheed Soomro last updated on 22/Nov/15

$$\mathcal{TH}an\mathfrak{K}\mathcal{S}for\mathcal{S}haring\mathcal{K}nowledge.Onethingmore$$$$\mathcal{I}wouldliketoask\underset{-}{howlnAisdefined?}$$$$\mathcal{D}oAandBneedtobeofsameorder?$$

Commented by Yozzi last updated on 22/Nov/15

$$IfAcanbediagonalised,$$$$tofindlnA,findtheeigenvector$$$$matrixVofthematrixA.With$$$$A^{{}^{\prime }}asadiaginalmatrixofthe$$$$eigenvaluesofAforitscorresponding$$$$Vwrite{A}^{{}^{\prime }}=V^{-1}AV(\ast )where{V}^{-1}isthe$$$$inversematrixofV.$$$$Suppose{A}^{{}^{\prime }}=\left[\begin{array}{c}a0\\ 0b\end{array}\right]\left(forexample\right).$$$$Then{lnA}^{{}^{\prime }}=\left[\begin{array}{c}lna0\\ 0lnb\end{array}\right].$$$$Takinglnonbothsidesof(\ast )$$$${lnA}^{{}^{\prime }}=V^{-1}\left(lnA\right)V\Rightarrow lnA=V\left({lnA}^{{}^{\prime }}\right)V^{-1}.$$

Commented by Rasheed Soomro last updated on 22/Nov/15

$${\mathcal{THANK}}^{\mathcal{S}sss}!$$

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