- Mathematics | Eigen Values and Eigen Vectors
- Matrix Multiplication
- Null Space and Nullity of a Matrix
For a given matrix A the set of all eigenvectors of A associated with an eigenvalue
Below are some useful properties of eigenvalues and eigenvectors in addition to the properties which are already listed in the article Mathematics | Eigen Values and Eigen Vectors.
Note: ker stands for Kernel which is another name for null space.
Computing Eigenvalues, Eigenvectors, and Eigenspaces:
Consider given 2 X 2 matrix:
Step 1: Characteristic polynomial and Eigenvalues. The characteristic polynomial is given by det( ) After we factorize the characteristic polynomial, we will get which gives eigenvalues as and Step 2: Eigenvectors and Eigenspaces We find the eigenvectors that correspond to these eigenvalues by looking at vectors x such that For we obtain After solving the above homogeneous system of equations, we will obtain a solution space This eigenspace is one dimensional as it possesses a single basis vector. Similarly, we find eigenvector for by solving the homogeneous system of equations This means any vector , where such as is an eigenvector with eigenvalue 2. This means eigenspace is given as
The two eigenspaces
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