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Tridiagonal Matrices are special square matrices. Their nonzero
elements only occure at diagonals and subdiagonals. Other
entries are zero. Such matrices play an important role in
matrix theory and matrix computation. This paper consider such
matrix as a function of its diagonal vector, that is, the
subdiagonals are fixed constants. If we consider the set of all
vectors such that the value of this function is a singular
matrix, then this set will be the union of n disjoint surfaces
in n dimensional space and each surface is a connected closed
set. We will explore the geometric properties of these
surfaces. Moreover, the set of all vectors such that the value
of the function is a nonsingular matrix and the geometric
meaning of eigenvalues will also be discussed in this paper.
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