A tridiagonal matrix is a square matrix whose nonzero entries can occur only on the main diagonal, the subdiagonal, and the superdiagonal,
![[a_(11) a_(12) 0 0 ... 0 0; a_(21) a_(22) a_(23) ... ... 0 0; 0 a_(32) a_(33) ... ... a_(n-2,n-1) 0; | ... ... ... ... a_(n-1,n-1) a_(n-1,n); 0 0 ... ... ... a_(n,n-1) a_(nn)].](/images/equations/TridiagonalMatrix/NumberedEquation1.svg) |
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Every tridiagonal matrix is both an upper and a lower Hessenberg matrix. If its diagonal entries are 
, its superdiagonal entries
are 
,
and its subdiagonal entries are 
, then the determinants 
of its leading 
principal submatrices satisfy the continuant
recurrence
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In particular, the determinant of the full matrix is 
and can be computed in linear time.
Computing the determinant of such a matrix requires only 
(as opposed to 
)
arithmetic operations (Acton 1990, p. 332). Efficient solution of the matrix
equation 
for 
,
where 
is a tridiagonal matrix, can be performed in the Wolfram
Language using LinearSolve
on 
,
represented as a SparseArray.
