Sum of Unitary Matrices Question

[RJLiberator]

Homework Statement

Find an example of two unitary matrices that when summed together are not unitary.

Homework Equations

The Attempt at a Solution

A = \begin{pmatrix}
0 & -i\\
i & 0\\
\end{pmatrix}

B = \begin{pmatrix}
0 & 1\\
1 & 0\\
\end{pmatrix}

A+B =
A = \begin{pmatrix}
0 & 1-i\\
1+i & 0\\
\end{pmatrix}

So we see that the hermitian conjugate of (A+B) is identical to A+B.

So (A+B)(A+B) =
A = \begin{pmatrix}
2 & 0\\
0 & 2\\
\end{pmatrix}

So since it is a diagonal matrix of 2, this is not the identity matrix. We can safely conclude that while A is unitary, B is unitary, (A+B) is NOT unitary.

Safe understanding?

Thanks

 

[Dick]

Homework Statement

Find an example of two unitary matrices that when summed together are not unitary.

Homework Equations

The Attempt at a Solution

A = \begin{pmatrix}
0 & -i\\
i & 0\\
\end{pmatrix}

B = \begin{pmatrix}
0 & 1\\
1 & 0\\
\end{pmatrix}

A+B =
A = \begin{pmatrix}
0 & 1-i\\
1+i & 0\\
\end{pmatrix}

So we see that the hermitian conjugate of (A+B) is identical to A+B.

So (A+B)(A+B) =
A = \begin{pmatrix}
2 & 0\\
0 & 2\\
\end{pmatrix}

So since it is a diagonal matrix of 2, this is not the identity matrix. We can safely conclude that while A is unitary, B is unitary, (A+B) is NOT unitary.

Safe understanding?

Thanks

That's fine. Even simpler, if $I$ is the identity matrix, then $I$ is unitary, so is $-I$. $I+(-I)=0$. $0$ is not unitary.