<s>[INST] Exponentiating Matrices - Am I doing it correctly? I recently learned how to exponentiate matrices by extending the power series of $e^x$ to matrices:
$$e^A=A^0+A^1+\frac{A^2}{2!}+\frac{A^3}{3!}+...$$
I thought that this was pretty cool, so I decided to try it with the matrix
$$A=\begin{pmatrix}1&1\\1&1\end{pmatrix}$$
I then realized that I would have to find a general formula for $A^n$. I ended up getting
$$A^n=\begin{pmatrix}2^{n-1}&2^{n-1}\\2^{n-1}&2^{n-1}\end{pmatrix}$$
for $n\ge 1$, and for $n=0$, I have that $A^0=I$. So my final answer would be in the form
$$e^A=\begin{pmatrix}\alpha&\alpha-1\\\alpha-1&\alpha\end{pmatrix}$$
where
$$\alpha=1+2^0+\frac{2^1}{2!}+\frac{2^2}{3!}+...$$
$$\alpha=\frac{1}{2}\bigg(1+2^0+2^1+\frac{2^2}{2!}+\frac{2^3}{3!}+...\bigg)$$
$$\alpha=\frac{1}{2}(e^2+1)$$
and so my answer would be
$$e^A=\begin{pmatrix}\frac{e^2+1}{2}&\frac{e^2-1}{2}\\\frac{e^2-1}{2}&\frac{e^2+1}{2}\end{pmatrix}$$
Is this correct?
 [/INST] Your method is correct, as you've been told. But here's another way of doing this that may interest you. Let$$M=\frac1{\sqrt2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}.$$Then$$M^{-1}=\frac1{\sqrt2}\begin{pmatrix}1&1\\-1&1\end{pmatrix}\text{ and }M^{-1}.A.M=\begin{pmatrix}2&0\\0&0\end{pmatrix}.$$Therefore$$M^{-1}.e^A.M=e^{\begin{pmatrix}2&0\\0&0\end{pmatrix}}=\begin{pmatrix}e^2&0\\0&1\end{pmatrix}.$$So,$$e^A=M.\begin{pmatrix}e^2&0\\0&1\end{pmatrix}.M^{-1}=\begin{pmatrix}\frac{e^2+1}{2}&\frac{e^2-1}{2}\\\frac{e^2-1}{2}&\frac{e^2+1}{2}\end{pmatrix}.$$
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