# Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.

Let A and B be Hermitian matrices. Answer true or false for each of the statements that follow. In each case, explain or prove your answer. The eigenvalues of AB are all real.
You can still ask an expert for help

• Questions are typically answered in as fast as 30 minutes

Solve your problem for the price of one coffee

• Math expert for every subject
• Pay only if we can solve it

oppturf
Step 1
The given statement is False
Step 2
Counter Example:
Let $A=\left[\begin{array}{cc}2& -2i\\ 2i& 0\end{array}\right]$ and $B=\left[\begin{array}{cc}2& 2i\\ -2i& 0\end{array}\right]$
A and B are Hermitian matrices
Now consider the product AB
$AB=\left[\begin{array}{cc}2& -2i\\ 2i& 0\end{array}\right]\left[\begin{array}{cc}2& 2i\\ -2i& 0\end{array}\right]=\left[\begin{array}{cc}4+4{i}^{2}& 4i-0\\ 4i-0& 4{i}^{2}\end{array}\right]$
$⇒AB=\left[\begin{array}{cc}4-4& 4i\\ 4i& -4\end{array}\right]=\left[\begin{array}{cc}0& 4i\\ 4i& -4\end{array}\right]$
Consider characteristic equation for the matrix AB to get eigenvalue
$det\left(A-\lambda I\right)=|\begin{array}{cc}0-\lambda & 4i\\ 4i& -4-\lambda \end{array}|=|\begin{array}{cc}-\lambda & 4i\\ 4i& -4-\lambda \end{array}|=0$
$⇒-\lambda \left(-4-\lambda \right)-\left(16{i}^{2}\right)=0$
$⇒4\lambda +-{\lambda }^{2}+16=0$
$⇒{\lambda }^{2}+4\lambda +16=0$
$⇒\lambda =\frac{-4±\sqrt{{4}^{2}-4\left(16\right)}}{2}$
$⇒\lambda =\frac{-4±\sqrt{16-64}}{2}$
$⇒\lambda =\frac{-4±\sqrt{48i}}{2}=-2±2\sqrt{3}i$
Clearly the eigenvalues are not real numbers.
Step 3
The given statement is False.
Jeffrey Jordon