Given a matrix equation, a vector equation and a scalar equation, equation is called an Eigenvalue and equation is called an Eigenvector if the following equation is satisfied:

    equation

That means, applying the linear transformation equation to equation (multiplying the matrix with the vector) gives the same result as multiplying the same vector by a scalar equation. In many cases, applying a linear transformation to a vector will give a vector that points into a different direction. Only in some cases, the new vector will have the same direction (even if the orientation is inverted) as the original vector. In this case the new vector can be obtained from the old one by multiplying it with a scalar.

The above equation has a trivial solution for equation. Typically, we want to find the Eigenvectors and Eigenvalues of a matrix with equation.

Finding the Eigenvalues of a matrix

Given a square matrix equation, we want to find its Eigenvalues equation for a nonzero vector equation. We transform the above equation as follows:

(1)   equation

If the matrix equation is invertible, then the solution is: equation. However, we want to find a solution for a equation. Hence, we can only find a solution if equation is not invertible. This is the case when the determinant of this matrix is zero:

    equation

The determinant equation gives us a polynomial. The roots of this polynomial are the Eigenvalues of equation.

Example

Let us find the Eigenvalues of the following matrix:

    equation

We find the polynomial given by the determinant equation:

(2)   equation

The Eigenvalues are the roots of this polynomial and can be found as follows:

(3)   equation

From this we can see that the Eigenvalues of equation are equation and equation.

Finding the Eigenvectors of a Matrix

To find the Eigenvectors equation of a matrix equation, we need to know its Eigenvalues equation and can find equation by solving equation.

Example

We continue with our example from above. Remember our matrix equation and Eigenvalues:

(4)   equation

We first find the Eigenvectors for the Eigenvalue equation:

(5)   equation

This linear system of equations is not independent. It is satisfied for any vector equation such that equation. The Eigenvectors of equation corresponding to the Eigenvalue equation are: equation for any equation.

We repeat the same process for equation and find the the Eigenvectors equation for any equation.