QR-factorization of a tridiagonal matrix super diagonals question I understand it is possible to QR-factorize a tridiagonal matrix A by performing Given's plane rotations:
$$ J(n-1,n)J(n-2,n-1)... J(1,2) A =R$$ where $R$ is upper triangular. I have read that in this case the first two super-diagonals of R will be non-zero. I am having a hard time visualizing this. I can see the given product gives an upper-triangular easily, but i struggle to see what happens above the diagonal. Any help would be greatly appreciated.
EDIT:
I have found now that $J(k-1,k) $ only affects rows $k-1$ and $k \space \space$, so if we can show $$J(2,3)J(1,2)A$$ has nonzero etc then we are done
EDIT 2: I have shown it this way. If anyone still wishes to answer then go ahead.

When working with Givens rotations on structured matrices, it is always instructive to draw a picture and of course to notice that pre-multiplying with $J(k,k+1)$ acts only on rows $k$ and $k+1$ (and post-multiplying affects the columns $k$ and $k+1$). In the pictures, $\times$ denotes nonzero entries, $\color{red}\times$ indicates the entries affected by the rotation, $\color{blue}\otimes$ marks the entry eliminated by the rotation, and $\color{red}\otimes$ shows where a new entry will be created:
$$
\begin{split}
    \underbrace{\begin{bmatrix}
        \times & \times &        &        &        &        \\
        \times & \times &        &        &        &        \\
               &        & 1      &        &        &        \\
               &        &        & 1      &        &        \\
               &        &        &        & 1      &        \\
               &        &        &        &        & 1
    \end{bmatrix}}_{J(1,2)}
    \underbrace{\begin{bmatrix}
        \color{red}\times & \color{red}\times & \color{red}\otimes       &        &        &        \\
        \color{blue}\otimes & \color{red}\times & \color{red}\times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{A}
    &=
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(1,2)A}
    \\
    \underbrace{\begin{bmatrix}
        1      &        &        &        &        &        \\
               & \times & \times &        &        &        \\
               & \times & \times &        &        &        \\
               &        &        & 1      &        &        \\
               &        &        &        & 1      &        \\
               &        &        &        &        & 1
    \end{bmatrix}}_{J(2,3)}
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \color{red}\times & \color{red}\times & \color{red}\otimes       &        &        \\
               & \color{blue}\otimes & \color{red}\times & \color{red}\times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(1,2)A}
    &=
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(2,3)J(1,2)A}
    \\
    \underbrace{\begin{bmatrix}
        1      &        &        &        &        &        \\
               & 1      &        &        &        &        \\
               &        & \times & \times &        &        \\
               &        & \times & \times &        &        \\
               &        &        &        & 1      &        \\
               &        &        &        &        & 1
    \end{bmatrix}}_{J(3,4)}
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \color{red}\times & \color{red}\times & \color{red}\otimes       &        \\
               &        & \color{blue}\otimes & \color{red}\times & \color{red}\times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(2,3)J(1,2)A}
    &=
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(3,4)J(2,3)J(1,2)A}
    \\
    \underbrace{\begin{bmatrix}
        1      &        &        &        &        &        \\
               & 1      &        &        &        &        \\
               &        & 1      &        &        &        \\
               &        &        & \times & \times &        \\
               &        &        & \times & \times &        \\
               &        &        &        &        & 1
    \end{bmatrix}}_{J(4,5)}
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \color{red}\times & \color{red}\times & \color{red}\otimes       \\
               &        &        & \color{blue}\otimes & \color{red}\times & \color{red}\times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(3,4)J(2,3)J(1,2)A}
    &=
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times \\
               &        &        &        & \times & \times 
    \end{bmatrix}}_{J(4,5)J(3,4)J(2,3)J(1,2)A}
    \\
    \underbrace{\begin{bmatrix}
        1      &        &        &        &        &        \\
               & 1      &        &        &        &        \\
               &        & 1      &        &        &        \\
               &        &        & 1      &        &        \\
               &        &        &        & \times & \times \\
               &        &        &        & \times & \times
    \end{bmatrix}}_{J(5,6)}
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \color{red}\times & \color{red}\times \\
               &        &        &        & \color{blue}\otimes & \color{red}\times 
    \end{bmatrix}}_{J(4,5)J(3,4)J(2,3)J(1,2)A}
    &=
    \underbrace{\begin{bmatrix}
        \times & \times & \times &        &        &        \\
               & \times & \times & \times &        &        \\
               &        & \times & \times & \times &        \\
               &        &        & \times & \times & \times \\
               &        &        &        & \times & \times \\
               &        &        &        &        & \times 
    \end{bmatrix}}_{J(5,6)J(4,5)J(3,4)J(2,3)J(1,2)A=R}
\end{split}
$$
