Editing Householder transformation (section)

==== Tridiagonalization (Hessenberg) ====
{{main|Tridiagonal matrix}}

{{see also|Hessenberg matrix#Householder transformations}}

This procedure is presented in Numerical Analysis by Burden and Faires, and works when the matrix is symmetric. In the non-symmetric case, it is still useful as a similar procedure can result in a Hessenberg matrix.

It uses a slightly altered <math>\operatorname{sgn}</math> function with <math>\operatorname{sgn}(0) = 1</math>.<ref name='burden'>{{cite book |last1=Burden |first1=Richard |last2=Faires |first2=Douglas |last3=Burden |first3=Annette  |title=Numerical analysis |date=2016 |publisher=Thomson Brooks/Cole |isbn=9781305253667 |edition=10th}}</ref>
In the first step, to form the Householder matrix in each step we need to determine <math display="inline">\alpha</math> and <math display="inline">r</math>, which are:
:<math>\begin{align}
  \alpha &= -\operatorname{sgn}\left(a_{21}\right)\sqrt{\sum_{j=2}^n a_{j1}^2}; \\
       r &= \sqrt{\frac{1}{2}\left(\alpha^2 - a_{21}\alpha\right)};
\end{align}</math>

From <math display="inline">\alpha</math> and <math display="inline">r</math>, construct vector <math display="inline">v</math>:

:<math>\vec v^{(1)} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix},</math>

where <math display="inline">v_1 = 0</math>, <math display="inline">v_2 = \frac{a_{21} - \alpha}{2r}</math>, and 
:<math>v_k = \frac{a_{k1}}{2r}</math> for each <math>k = 3, 4 \ldots n</math>

Then compute:
:<math>\begin{align}
      P^1 &= I - 2\vec v^{(1)} \left(\vec v^{(1)}\right)^\textsf{T} \\
  A^{(2)} &= P^1 AP^1
\end{align}</math>

Having found <math display="inline">P^1</math> and computed <math display="inline">A^{(2)}</math> the process is repeated for <math display="inline">k = 2, 3, \ldots, n - 2</math> as follows:

:<math>\begin{align}
     \alpha &= -\operatorname{sgn}\left(a^k_{k+1,k}\right)\sqrt{\sum_{j=k+1}^n \left(a^k_{jk}\right)^2} \\[2pt]
          r &= \sqrt{\frac{1}{2}\left(\alpha^2 - a^k_{k+1,k}\alpha\right)} \\[2pt]
      v^k_1 &= v^k_2 = \cdots = v^k_k = 0 \\[2pt]
  v^k_{k+1} &= \frac{a^k_{k+1,k} - \alpha}{2r} \\
      v^k_j &= \frac{a^k_{jk}}{2r} \text{ for } j = k + 2,\ k + 3,\ \ldots,\ n \\
        P^k &= I - 2\vec v^{(k)} \left(\vec v^{(k)}\right)^\textsf{T} \\
  A^{(k+1)} &= P^k A^{(k)}P^k
\end{align}</math>

Continuing in this manner, the tridiagonal and symmetric matrix is formed.