Homogeneous Algebraic Equations and the Characteristic-Value Problem

9 Homogeneous Algebraic Equations and THE Characteristic-Value Problem 

A special class of homogeneous linear algebraic equations arises in the study of vibrating systems, structure analysis, and electric circuit system analysis, and in the solution and stability analysis of linear ordinary differential equations (Chap. 5). This system of equations has the form 

Before we proceed with developing methods of solution, we examine Eq. (2.147) from a geometric perspective. The multiplication of a vector by a matrix is a bnear transformation of the original vector to a new vector of different direction and length. For example, matrix A transforms the vector y to the vector z in the operation 

In contrast to this, if x is the eigenvector of A, then the multiplication of the eigenvector x by matrix A yields the same vector x multiplied by a scalar k, that is, the same vector but of different length  

It can be stated that for a nonsingular matrix A of order n, there are n characteristic directions in which the operation by A does not change the direction of the vector, but only changes its length. More simply stated, matrix A has n eigenvectors and n eigenvalues. The types of eigenvalues that exist for a set of special matrices are listed in Table 2.4. 

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Homogeneous Algebraic Equations and the Characteristic-Value Problem 121... 

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