Tridiagonal matrix: Difference between revisions

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As a side note, an '''unreduced''' symmetric tridiagonal matrix is a matrix containing non-zero off-diagonal elements of the tridiagonal, where the eigenvalues are distinct while the eigenvectors are unique up to a scale factor and are mutually orthogonal.<ref>{{cite book |last1=Dhillon |first1=Inderjit Singh |title=A New O(n 2 ) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem |page=8 |url=http://www.cs.utexas.edu/~inderjit/public_papers/thesis.pdf}}</ref>

=== Similarity to symmetric tridiagonal matrix ===

Given a given real tridiagonal, ''unsymmetic'' 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 the product of off-diagonal entries is ''strictly'' positive <math>b_i c_i > 0 </math> and define a transformation matrix <math>D</math> by

:<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>J:=D^{-1} T D </math> yields a ''symmetric''<ref>{{Cite web|url=http://www.math.hkbu.edu.hk/ICM/LecturesAndSeminars/08OctMaterials/1/Slide3.pdf|title=www.math.hkbu.edu.hk math lecture|last=|first=|date=|website=|url-status=live|archive-url=|archive-date=|access-date=}}</ref> tridiagonal matrix <math>J</math> by

:<math>

J:=D^{-1} T D

= \begin{pmatrix}

a_1 & \sqrt{b_1 c_1} \\

\sqrt{b_1 c_1} & a_2 & \sqrt{b_2 c_2} \\

& c_2 & \ddots & \ddots \\

& & \ddots & \ddots & \sqrt{b_{n-1} c_{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.

==Computer programming==