Characteristic Equation, Eigenvalues

 

The characteristic equation of a square matrix A is:

where λ is a scaling factor.

This returns a polynomial, which is called the characteristic polynomial:

The roots of this polynomial are called Eigenvalues.

Consider the following matrix:

The characteristic polynomial is:

The roots of this polynomial are the eigenvalues:

 Also, the Cayley-Hamilton rule, states that a square matrix must satisfy its own characteristic polynomial. So, for the previous example of matrix A:

The eigenvalues answer the question, which set of vectors v, called eigenvectors of the matrix M, multiplied by the matrix M (i.e. a transformation matrix), doesn’t change the elements of the vectors v, but they just scale vector’s magnitude?

For the matrix A of the previous example, with eigenvalues of 5 and -2, a set of eigenvectors could be:

Because:

So the vectors multiplied by the square matrix are just scaled by a factor of 5.