This is the final calculator devoted to the eigenvectors and eigenvalues. The first one was the Characteristic polynomial calculator, which produces a characteristic equation suitable for further processing. Second calculator - the Eigenvalue calculator solves that equation to find eigenvalues (using analytical methods, that's why it works only up to 4th degree), and the calculator below calculates eigenvectors for each eigenvalue found. Some theories can be found below the calculator.
How to find Eigenvectors
Let me repeat the definition of eigenvectors and eigenvalues from the Eigenvalue calculator.
There are vectors for which matrix transformation produces the vector that is parallel to the original vector.
where is some scalar number.
These vectors are called the eigenvectors of A, and these numbers are called the eigenvalues of A.
We use the following form of the equation above: , where I is the identity matrix, to find the eigenvalues by solving the characteristic equation
After we found the eigenvalues, we can find eigenvectors. We should plug each concrete eigenvalue into the equation and solve it for v. This means that we simply need to solve the following system of linear equations (in matrix form):
This is a homogeneous system of linear equations, and even more, its equations are NOT independent. That is, the system has infinitely many solutions. This is because we have a family of eigenvectors (including zero vector), or eigenspace, for each eigenvalue. So, when you are asked to find eigenvectors for the matrix, you really need to pick up some "beautiful" solution for a system of linear equations obtained for each eigenvalue, that is, some sample eigenvector with possible no fractions and small positive integers.
In most cases, eigenvalue produces a homogeneous system with one independent variable. However, some cases have eigenvalue with multiplicity more than 1 (f.e. in case of double roots). In such cases, a homogeneous system will have more than one independent variable, and you will have several linearly independent eigenvectors associated with such eigenvalue - one for each independent variable.