Tridiagonal matrix

Remarks

Example: $\begin{pmatrix} 2 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 2 \end{pmatrix}$ is tridiagonal, because the only entries with $\abs{i - j} > 1$ are $a_{13}$ and $a_{31}$, and both are $0$. Non-example: $\begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ is not tridiagonal, because $a_{13} = 1$ and $\abs{1 - 3} = 2$. A tridiagonal matrix is often written with diagonal $a_1, \dots, a_n$, superdiagonal $b_1, \dots, b_{n-1}$ and subdiagonal $c_1, \dots, c_{n-1}$. Then the leading principal minors $f_k$ satisfy $f_k = a_k f_{k-1} - b_{k-1} c_{k-1} f_{k-2}$, with $f_0 = 1$ and $f_{-1} = 0$. Equivalently, $A$ is tridiagonal when it is both upper and lower Hessenberg.