Characteristics of Eigenvectors and Eigenvalues

Let us define orthogonal unit vectors  in the linear vector space and unit vector set . These orthogonal vectors will form its basis (main components). Here .  

Let us find decomposition on the basis:

Therefore, - is the representation on basis. Then,:

.

Every line of matrix determines the projections of onto direction . Let us find out the angular ratios (cosines) between all the unit vectors , assigned by matrix.

It is evident that . Since the angles between vectors don't depend on the choice of coordinate system, . Consequently,  , and - is an identity matrix of dimension .

Let us find the product :

Since , then diagonally in matrix received there are square sums of all vectors' projections on the direction :. Since all , then  .

       Let us take an identity:

.

Let us separate the  -th column-vector from the left product:, where is a column of matrix , as well as from the right product:

Thus, ,and is an matrix eigenvector, which belongs to eigenvalue and the obtained equality determines a quadric surface:.  Therefore, eigenvectors can be established by solving the following characteristic equation:

 или:

For the system to have a nontrivial solution, it is necessary that the system determinant be , and therefore, will be evaluated by solving an -degree equation:

It is usually stipulated that . If we substitute a concrete eigenvalue into the system, we can obtain the relevant eigenvector .

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