Characteristic Polynomial
Calculating Eigenvalues
Calculating EigenvectorsHow to Examples Exercise
How to:
We originally said that eigenvalues are scalars which satisfy the equation:
A*v = lambda*v
Where A is our original matrix and lambda is one of our eigenvalues. However, we never specified what v was.
Our equation above does not work for all vectors, but only a small set of vectors known as eigenvectors.
To find a particular eigenvector, we need to compute the kernal of lambda, our eigenvalue, times the identity matrix minus A, or:
ker(lambda*eye(n) - A)
As always, MATLAB wouldn't be much of a program if it made us work this hard. The simplfied method is to input:
[V, D] = eig(A)
Here, D is an NxN matrix with our eigenvalues on its diagonal and V is an NxN matrix with columns that represent eigenvectors. So, their values satisfy the equation:
A*V = V*D
Or, if we want to look at one eigenvalue/eigenvector at a time:
A*V(i, :) = D(i, i)*V(i, :)
This stands for A times the i-th column of V (an eigenvector) equals the i-th row/i-th column entry of D (which of course falls on the diagonal), time the column of V.
Examples:
Exercise: