I have the following matrix :
$$ \mathbf A =\begin{bmatrix} 100 & 0 \\ 0 & 1 \\ \end{bmatrix}$$
I have to compute $ \mathbf A^{-1/2}$.
So I need spectral decomposition, $$ \mathbf A = \mathbf P \mathbf \Lambda\mathbf P',$$
$\mathbf P$ be a matrix with normalized eigenvectors and $\mathbf \Lambda$ is a diagonal matrix with diagonal elements be eigenvalues.
Eigenvalues of $ \mathbf A$ is $100$ and $1$.
But I stumbled to calculate eigenvector.
The characteristic equation is :
$$ \begin{bmatrix} 100 & 0 \\ 0 & 1 \\ \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix}= 100\begin{bmatrix} x_1 \\ x_2 \\ \end{bmatrix}$$
$$\Rightarrow 100x_1 = 100x_1$$
$$x_2 = 100 x_2$$
How is $x_2 = 100 x_2$ possible ?
And is there a simpler way to calculate any power of a diagonal matrix , for example, $ \mathbf A^{-1/2}$ ?