= Tridiagonal matrix
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= Tridiagonal
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A tridiagonal <matrix> has zero entries outside its main diagonal and the two adjacent diagonals. With nonzero elimination <pivots>, the <Thomas algorithm> solves its <linear system> in linear time by specialized <Gaussian elimination> and back substitution.
Nonsingularity alone does not ensure that the unpivoted <Thomas algorithm> is defined: the <invertible matrix> $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$ has a zero first <pivot>. Conditions such as <strict diagonal dominance> guarantee suitable nonzero <pivots>; otherwise <row interchanges> or another factorization are needed.
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