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We give an explicit formula for the solution to the initial value problem ofthe full symmetric Toda hierarchy. The formula is obtained by theorthogonalization procedure of Szeg\"{o}, and is also interpreted as aconsequence of the QR factorization method of Symes \cite{symes}. The sortingproperty of the dynamics is also proved for the case of a generic symmetricmatrix in the sense described in the text, and generalizations of tridiagonalformulae are given for the case of matrices with $2M+1$ nonzero diagonals.Comment: 13 pages, Latex

Explicit integration of the full symmetric Toda hierarchy and the sorting property