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