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## Homework Statement

Find the spectral decompostion of

$$

A=

\begin{matrix}

1 & 0 & 0\\

-1 & 1 & 1 \\

-1 & 0 & 2

\end{matrix}

$$

and use this to find $$ 2^{A} $$

## Homework Equations

## The Attempt at a Solution

I have found the eigenvalues to be : $$ \lambda_{1}=1 \text{ and } \lambda_{2}=2 $$

My textbook (matrices and linear transformations, Cullen) uses lagrange polynomials to find the spectral projectors:

$$

h_{1}(x)=\frac{x-2}{1-2}=-(x-2)\\

h_{2}(x)=\frac{x-1}{2-1}=(x-2)\\

E_{1}=h_{1}(A)\\

E_{2}=h_{2}(A)\\

\therefore A=E_{1}+2E_{2}

$$

However I now need to use this to solve for $$ 2^{A} $$

I think I need to write $$ 2^{x} $$ as an infinite series, but I am not sure how to do this.