- #1

- 163

- 0

## Homework Statement

The matrix [tex]C[/tex] is self-adjoint and positive definite on [tex]\mathbb{ C}^2[/tex]. Determine [tex]\sqrt{\mathbb{C}} \in \mathbb{ C}^{2x2} [/tex].

[tex]C=\left(\begin{array}{cc}5&-4i\\4i&5\end{array}\right)[/tex]

## The Attempt at a Solution

__Characteristic polynomial:__

[tex](5-\lambda)^2 - 16 = 0[/tex]

What I have done so far is solve for lambda, giving me 9 and 3 respectively. But then I don't really know what to do with those eigenvalues. Do I have to search the eigenvectors as well? In that case, what should I do with them?

I know that at the end I should get something in the following form to get the square root of the matrix:

[tex]\sqrt{C} = a\cdot\left(\begin{array}{cc}5&-4i\\4i&5\end{array}\right) + b\cdot \left(\begin{array}{cc}1&0\\0&1\end{array}\right)[/tex]

But I don't know how to figure out both a and b.

Any help will be appreciated. I have been looking for tutorials in google, but I couldn't find any regarding the procedure for the square roots of the matrix.