Editing Tridiagonal matrix (section)

=== Similarity to symmetric tridiagonal matrix ===
For ''unsymmetric'' or ''nonsymmetric'' tridiagonal matrices one can compute the eigendecomposition using a similarity transformation.
Given a real tridiagonal, nonsymmetric matrix 
:<math>
T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
& & \ddots & \ddots & b_{n-1} \\
& & & c_{n-1} & a_n
\end{pmatrix}
</math>
where <math>b_i \neq c_i </math>.
Assume that each product of off-diagonal entries is {{em|strictly}} positive <math>b_i c_i > 0 </math> and define a transformation matrix <math>D</math> by<ref name=kreer_operator>{{Cite journal | last1 = Kreer | first1 = M. | title = Analytic birth-death processes: a Hilbert space approach | doi = 10.1016/0304-4149(94)90112-0 | journal = Stochastic Processes and Their Applications | volume = 49 | issue = 1 | pages = 65–74 | year = 1994 }}</ref> 
:<math>
D := \operatorname{diag}(\delta_1 , \dots, \delta_n) 
\quad \text{for} \quad
\delta_i :=
\begin{cases}
1 & , \, i=1
\\
\sqrt{\frac{c_{i-1} \dots c_1}{b_{i-1} \dots b_1}} & , \, i=2,\dots,n \,.
\end{cases}
</math>

The [[Matrix_similarity|similarity transformation]] <math>D^{-1} T D </math> yields a ''symmetric'' tridiagonal matrix <math>J</math> by:<ref>{{cite web |first=Gérard |last=Meurant |title=Tridiagonal matrices |date=October 2008 |url=http://www.math.hkbu.edu.hk/ICM/LecturesAndSeminars/08OctMaterials/1/Slide3.pdf |via=Institute for Computational Mathematics, Hong Kong Baptist University}}</ref><ref name=kreer_operator/> 
:<math>
J:=D^{-1} T D 
= \begin{pmatrix}
a_1 & \sgn b_1 \,  \sqrt{b_1 c_1} \\
\sgn b_1 \, \sqrt{b_1 c_1} & a_2 & \sgn b_2 \, \sqrt{b_2 c_2} \\
& \sgn b_2 \,  \sqrt{b_2 c_2} & \ddots & \ddots \\
& & \ddots & \ddots & \sgn b_{n-1} \, \sqrt{b_{n-1} c_{n-1}} \\
& & & \sgn b_{n-1} \,  \sqrt{b_{n-1} c_{n-1}} & a_n
\end{pmatrix} \,.
</math>
Note that <math>T</math> and <math>J</math> have the same eigenvalues.