Tridiagonal matrix: Difference between revisions

Revision as of 05:10, 29 August 2024

Revision as of 06:08, 29 August 2024

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:<math> a - 2 \sqrt{bc} \cos \left (\frac{k\pi}{n+1} \right ), \qquad k=1, \ldots, n. </math>

A real [[symmetric matrix|symmetric]] tridiagonal matrix has real eigenvalues, and all the eigenvalues are [[Eigenvalues and eigenvectors#Algebraic multiplicity|distinct (simple)]] if all off-diagonal elements are nonzero.<ref>{{Cite book | last1 = Parlett | first1 = B.N. | title = The Symmetric Eigenvalue Problem |orig-year = 1980 | publisher =SIAM |date=1997 |oclc= 228147822 |series=Classics in applied mathematics |volume=20 |isbn={{Format ISBN|0898714028}}0-89871-402-8 }}</ref> Numerous methods exist for the numerical computation of the eigenvalues of a real symmetric tridiagonal matrix to arbitrary finite precision, typically requiring <math>O(n^2)</math> operations for a matrix of size <math>n\times n</math>, although fast algorithms exist which (without parallel computation) require only <math>O(n\log n)</math>.<ref>{{Cite journal |last1 = Coakley |first1= E.S. |last2=Rokhlin | first2=V. | title =A fast divide-and-conquer algorithm for computing the spectra of real symmetric tridiagonal matrices | doi = 10.1016/j.acha.2012.06.003 |journal = [[Applied and Computational Harmonic Analysis]] |volume = 34 |issue = 3 |pages = 379–414 |year =2012 |doi-access = free }}</ref>

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 thesis |last1=Dhillon |first1=Inderjit Singh |title=A New O(n²) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem |page=8 |type=PhD |publisher=University of California, Berkeley |id=CSD-97-971, ADA637073 |date=1997 |url=http://www.cs.utexas.edu/~inderjit/public_papers/thesis.pdf}}</ref>