- #1

Adgorn

- 130

- 18

## Homework Statement

Prove that a 2x2 complex matrix ##A=\begin{bmatrix} a & b \\

c & d\end{bmatrix}## is positive if and only if (i) ##A=A*##. and (ii) ##a, d## and ##\left| A \right| = ad-bc##

## Homework Equations

N/A

## The Attempt at a Solution

I got stuck at the first part. if ##A## is positive then by definition ##A=S^*S## for some matrix ##S## and thus ##A^*=(S^*S)^*=S^*S=A## and so ##A=A^*##.

However I cannot prove the opposite, that if ##A=A^*## then A is positive. Since ##A=A^*## A is normal and thus there exists an orthonormal basis of ##C^2## comprised of eigenvectors of A, say, {##v_1.v_2##}. Furthermore since ##A## is self-adjoint all eigenvalues of ##A## are real. If I could prove the eigenvalues of ##A## are also nonnegative the proof will be complete, but I cannot seem to manage to prove that and it might not even be true.

Regarding the second part, assuming I have proven the first part it is easy to show how ##a##, ##d## and ##ad-bc## must be real. However I can't seem to figure out how to prove they are nonnegative.

Help would be appriciated.