Page "Tridiagonal matrix" Paragraph 5

A tridiagonal matrix is of Hessenberg type ; in particular, a tridiagonal matrix is a direct sum of p 1-by-1 and q 2-by-2 matrices such that p + q / 2 = n -- the dimension of the tridiagonal.

Although a general tridiagonal matrix is not necessarily symmetric or Hermitian, many of those that arise when solving linear algebra problems have one of these properties.

Furthermore, if a real tridiagonal matrix A satisfies a < sub > k, k + 1 </ sub > a < sub > k + 1, k </ sub > > 0, so that the signs of its entries are symmetric, then it is similar to a Hermitian matrix, and hence, its eigenvalues are real.

The latter conclusion continues to hold if we replace the condition a < sub > k, k + 1 </ sub > a < sub > k + 1, k </ sub > > 0 by a < sub > k, k + 1 </ sub > a < sub > k + 1, k </ sub > ≥ 0.