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When a tridiagonal matrix is also [[Toeplitz matrix|Toeplitz]], there is a simple closed-form solution for its eigenvalues, namely:<ref>{{Cite journal | doi = 10.1002/nla.1811| title = Tridiagonal Toeplitz matrices: Properties and novel applications| journal = Numerical Linear Algebra with Applications| volume = 20| issue = 2| pages = 302| year = 2013| last1 = Noschese | first1 = S. | last2 = Pasquini | first2 = L. | last3 = Reichel | first3 = L. }}</ref><ref>This can also be written as <math> a + 2 \sqrt{bc} \cos(k \pi / {(n+1)}) </math> because <math> \cos(x) = -\cos(\pi-x) </math>, as is done in: {{Cite journal | last1 = Kulkarni | first1 = D. | last2 = Schmidt | first2 = D. | last3 = Tsui | first3 = S. K. | title = Eigenvalues of tridiagonal pseudo-Toeplitz matrices | doi = 10.1016/S0024-3795(99)00114-7 | journal = Linear Algebra and its Applications | volume = 297 | pages = 63 | year = 1999 | url = https://hal.archives-ouvertes.fr/hal-01461924/file/KST.pdf }}</ref>
:<math> a
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 | year = 1980 | publisher = Prentice Hall, Inc. }}</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>
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