Why I always get the different sign of non-diagonal elements compare to the text book(all values are same) For example: $$ \begin{bmatrix} 1 & 1 & 0 \\ 2 & 3 & 1 \\ 2 & 1 & 0 \\ \end{bmatrix} $$
To calculate $H_1$ $$H_1=I-2P$$ $$P=\frac{VV^T}{V^TV}$$ $$V=W-X$$
$W=\sqrt{2^2+2^2}$
$X=a_{12},a_{13}$
$V=\begin{bmatrix}\sqrt{8}-2\\-2\end{bmatrix}$
Hessenberg form:$H_1A_0H_1$
My answer:$\begin{bmatrix}1&\frac{1}{\sqrt2}&\frac{1}{\sqrt2}\\\sqrt8&\frac{5}{2}&\frac{3}{2}\\0&\frac{3}{2}&\frac{1}{2} \end{bmatrix}$
Textbook:$\begin{bmatrix}1&-\frac{1}{\sqrt2}&-\frac{1}{\sqrt2}\\-\sqrt8&\frac{5}{2}&\frac{3}{2}\\0&\frac{3}{2}&\frac{1}{2} \end{bmatrix}$
is it significant if i put my answer to calculate iegen value with QR alogrithm
if Yes how to correct the right answer?
if no what cause the different sign?
I have searched on google several times different sign means the decomposition is not unique