Eigenvalues and Eigenvectors

Given the matrix and the vectors and :

The vector is an eigenvector of with eigenvalue if:

In this case the eigenvalue is equal to as the resulting matrix is directly equal to .

In this case the eigenvector of has a value of 2 as the resulting vector is not directly equal to but is instead scaled by a factor of 2.

The eigenvalue and eigenvector together make an eigenpair.

Computing Eigenpairs

Solve for :

Where is the determinant, is the matrix, are the eigenvalues, and is the identity matrix.

Once the eigenvalues are computed you can use the following equation to compute the eigenvector:

Where is the eigenvector.

See below for more information:

When using eigenpairs with a linear system of differential equations the eigenvectors of matrix will pass through the origin of the xy graph, where x and y are given by the following:

When at an equilibrium point, where some causes to equal 0, as is the derivative of y, it is the slope. Thus:

As is a function of x and y and they are equal to 0 thus they must go though the origin at some point. #flag-review - expand/prove?