Polar Form of a Matrix (functional calculus )

Question

Let $T= \begin{pmatrix} 1 & 1 \\ 0 & 0 \\ \end{pmatrix} $.

Find the polar form of $T= W|T|$, where $|T|= \sqrt{T^*T}$, and $W$ is unitary.

I was able to find

$|T|= \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \end{pmatrix} $.

But I'm having trouble finding $W$, since $|T|$ here is not invertible.

Any help will be appreciated.

Answer 1

The matrix $$ W = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} $$ solves your equation.