Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In linear algebra, a tridiagonal matrix is a matrix that is "almost" a diagonal matrix. To be exact: a tridiagonal matrix has nonzero elements only in the main diagonal, the first diagonal below this, and the first diagonal above the main diagonal. A tridiagonal matrix is of Hessenberg type. 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 ak,k+1 ak+1,k > 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 ak,k+1 ak+1,k > 0 by ak,k+1 ak+1,k >= 0. The set of all n x n tridiagonal matrices form a 3n-2 dimensional vector space.

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