Now we will construct a so-called oscillation matrix, and in  it is shown that all the properties mentioned above are strictly connected to the properties of the oscillation matrix. Suppose our oscillation system consists of masses , located at points . Note that the masses are not fixed but movable in the direction of the -axis. With we denote the so-called influence function. It denotes the deflection at point under the influence of a unit force at point . Denote . Assume to be the deflection of these masses, under the influence of a force:
As already mentioned, in  Krein observed that the oscillation properties, given above, are closely related to the influence coefficients . Even more, the oscillation properties are related to the fact that all minors (a minor is the determinant of a submatrix of the given matrix) of the matrix (of all orders) have to be nonnegative. The theory connected to these matrices is called the theory of oscillation matrices.
Important properties of these oscillation matrices, and their connections with the oscillation properties where investigated in the papers [80,81,82]. Also other authors show their interest in this field of matrices [64,75,76], especially the article  contains several references related to positive and nonnegative matrices. The choice of the term oscillation matrix by the authors of the book  comes from the following circumstance: (citation from , Introduction, page 3)
As soon as for a finite system of points, the matrix of coefficients of influence of a given linear elastic continuum is an oscillation matrix (as it always is in the case for a string or a rod supported at the endpoints in the usual manner), this automatically implies the oscillation properties of the vibration of the continuum, for any distribution of masses at these points.
The theory of oscillation matrices (sometimes also other matrix theories), has an analogue in the theory of integral equations. For the oscillation matrices this corresponds to the theory of the following integral equation:
It follows from the definition that an oscillation kernel is characterized by the following inequalities:
For the purpose of our thesis this introduction into the theory of oscillation matrices is enough. In the remainder of this section we will take a closer look at two types of oscillation matrices: Jacobi matrices and one-pair matrices. Jacobi matrices are a special type of tridiagonal matrices, namely irreducible ones, and one-pair matrices are a special type of semiseparable matrices, namely symmetric generator representable semiseparable matrices of semiseparability rank . Also attention will be paid to the most important properties of these matrices, which are of interest in the theory of oscillation matrices.