I have to find the givens rotation matrix that will transform the following vector $[1, 1, -1]^T$ to $[y, 1, 0]^T$ (basically to insert a $0$ on the third position without altering the second one).
I tried to solve this problem but I'm not sure I am correct. My approach was the following:
$$\begin{bmatrix} c & 0 & -s\\ 0 & 1 & 0 \\ s & 0 & c \end{bmatrix}\cdot \begin{bmatrix}1\\1\\-1 \end{bmatrix}= \begin{bmatrix} y\\1\\0 \end{bmatrix}$$
So I have the following equation set:
$$c - s = y \text{ and }s - c = 0 $$
Which means that s = c so, theta angle is 45 degrees meaning that the Givens matrix for that rotation its the following:
$$\begin{bmatrix} √2/2 & 0 & -√2/2\\ 0 & 1 & 0 \\ √2/2 & 0 & √2/2 \end{bmatrix}$$
Am I right?
Yes, I was right, the matrix is a correct Givens Rotation Matrix that respects the required transformation.