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First write $T=\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)$ with $a,d>0$. Direct multiplication givesThe inner product of its two columns isIf $E$ is orthogonal this is zero, and positivity of the bracket forces $b=0$. The first column must then have length one, so $d/a=1$. Thus $T=aI$, $E=I$, and $J_{1}T=T{J}_1$.

For the general QR decomposition, apply the Gram-Schmidt process to the ordered columns $v_1,\dots ,v_N$ of the invertible real matrix $A$. At stage $j$, subtract projections onto the already chosen orthonormal vectors to get $u_j$; independence of the columns gives $u_j\ne 0$. Set $q_j=u_j/\Vert u_j\Vert$. The matrix $B$ with columns $q_j$ is orthogonal, andis upper triangular. Its diagonal entry $C_{jj}=\Vert u_j\Vert$ is strictly positive. This proves $A=BC$ with the required properties.

Now suppose $KA=AK$ in dimension $2n$. Substitution of $A=BC$ givesThe matrix $K$ is orthogonal, as are $B$ and $B^{-1}$, so this matrix and $E=K^{-1}CK{C}^{-1}$ are orthogonal. Regard $C$ as upper triangular in $2\times 2$ blocks, with invertible diagonal blocks $C_i$. The inverse is upper triangular in the same blocks, by solving the triangular block equations; multiplication by block diagonal $K$ preserves this pattern. Thus $E$ is block upper triangular.

An orthogonal block upper-triangular matrix is block diagonal. Indeed its first two columns are supported in the first two rows and are orthonormal, so span that first coordinate plane. Orthogonality of every subsequent column to them makes its first two entries zero. The remaining lower-right matrix is orthogonal; iterate this argument. Therefore $E=\mathrm{diag}(E_1,\dots ,E_n)$ with each $E_i$ orthogonal. The rule for diagonal blocks of products of block upper-triangular matrices givesEach $C_i$ is itself an upper-triangular real $2\times 2$ matrix with positive diagonal. The first calculation gives $E_i=I$. Consequently $E=I$, whence $CK=KC$. Finally $B=A{C}^{-1}$ is a product of matrices commuting with $K$, so $BK=KB$. We have provedThis is positive-diagonal QR decomposition preserves a complex structure; the positivity convention is important in the two-dimensional step.