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.
 A: 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}
$$
A: As you have noticed in your first edit, a single Givens rotation will affect only two rows. Since you apply it only on neighboring rows corresponding to subdiagonal elements, each will affect rows $k$ and $k+1$ for some $k \in \{1,2,\dots,m-1\}$.
Now, the only possibly non-zero elements in the $i$-th row of $A = [ a_{ij} ]$ are $a_{i,i-1}, a_{ii}, a_{i,i+1}$ (disregarding obvious exceptions of the first and the last row, as they don't affect the statement).
So, when we compute the linear combination of the $k$-th (for which we have already achieved $a_{k-1,k} = 0$) and $(k+1)$-st rows, induced by premultiplying $A$ with a Givens rotation defined by
$$\begin{bmatrix} c & s \\ -s & c \end{bmatrix},$$
we get
$$\begin{bmatrix}
0 & \cdots & 0 & \color{blue}{c a_{k,k} + s a_{k+1,k}} & c a_{k,k+1} + s a_{k+1,k+1} & \color{red}{c a_{k,k+2} + s a_{k+1,k+2}} & 0 & \cdots & 0 \\
0 & \cdots & 0 & -s a_{k,k} + c a_{k+1,k} & \color{blue}{-s a_{k,k+1} + c a_{k+1,k+1}} & -s a_{k,k+2} + c a_{k+1,k+2} &0 & \cdots & 0
\end{bmatrix}$$
The diagonal elements of the new matrix are painted blue, so it is clear that we have introduced a potentially non-zero element
$$\color{red}{c a_{k,k+2} + s a_{k+1,k+2}} = s a_{k+1,k+2}$$
at the position $(k, k+2)$, thus developing additional superdiagonal. Furthermore, the elements right to it are still equal to zero and will remain such, because we're done with the $k$-th row. Also, we have introduced no non-zeroes to the $(k+1)$-st row, so the next step can incorporate the same logic we have just described.
