Eigenvectors and Eigenvalues

For a matrix A, if there is vector x and value λ, such that it holds:

then x is called its eigenvector and λ is called its eigenvalue.

It is said that vector x is in the same direction as Ax (which is exceptional property). The parameter λ expresses whether and how much the vector shrinks or stretches.

It follows that multiplying the eigenvector by power of λ^p is like multiplying it p times by the matrix A:

There may exist multiple eigenvectors and eigenvalues. All vectors x can be found by solving equation:

This equation has a non-zero solution only if the determinant of the matrix is zero. In according, the parameters λ can be found by solving equation:

For example the two eigenvectors and eigenvalues for the 2x2 matrix:

Additional properties of nxn matrix A are the trace (sum of diagonal elements) which is sum of eigenvalues:

and the determinant which is product of eigenvalues: