Editing Hessenberg matrix (section)

===Jacobi (Givens) rotations===
A [[Jacobi rotation]] (also called Givens rotation) is an orthogonal matrix transformation in the form
:<math>
A\to A'=J(p,q,\theta)^TAJ(p,q,\theta) \;,
</math>
where <math>J(p,q,\theta)</math>, <math>p< q</math>, is the Jacobi rotation matrix with all matrix elements equal zero except for
::<math>\left\{\begin{align}
 J(p,q,\theta)_{ii} &{}= 1 \; \forall i \ne p,q\\
 J(p,q,\theta)_{pp} &{}= \cos(\theta) \\
 J(p,q,\theta)_{qq} &{}= \cos(\theta) \\
 J(p,q,\theta)_{pq} &{}= \sin(\theta) \\
 J(p,q,\theta)_{qp} &{}= -\sin(\theta) \;.
\end{align}\right.</math>
One can zero the matrix element <math>A'_{p-1,q}</math> by choosing
the rotation angle <math>\theta</math> to satisfy the equation
:<math>
A_{p-1,p}\sin\theta+A_{p-1,q}\cos\theta=0 \;,
</math>
Now, the sequence of such Jacobi rotations with the following <math>(p,q)</math>
:<math>
(p,q)=(2,3),(2,4),\dots,(2,n),(3,4),\dots,(3,n),\dots,(n-1,n)
</math>
reduces the matrix <math>A</math> to the lower Hessenberg form.<ref>{{cite journal | arxiv=1501.07812 | doi=10.1016/j.laa.2015.08.026 | title=Quasiseparable Hessenberg reduction of real diagonal plus low rank matrices and applications | date=2016 | last1=Bini | first1=Dario A. | last2=Robol | first2=Leonardo | journal=Linear Algebra and Its Applications | volume=502 | pages=186–213 }}</ref>