Tridiagonal matrix: Difference between revisions

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===Eigenvalues===

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