Editorial Staff
Eigenvectors play a crucial role in many areas of mathematics and physics. Understanding their calculation is essential for various applications.
Let’s consider a square matrix $A$ of size $n \times n$.
Step 1: Find Eigenvalues
Start by finding the eigenvalues of matrix $A$ using the eigenvalue calculation method explained in the previous section.
See – How to Calculate Eigenvalues?
Step 2: Substitute Eigenvalues
For each eigenvalue $\lambda_i$, substitute it back into the equation $A \mathbf{x} = \lambda \mathbf{x}$, where $\mathbf{x}$ is the eigenvector.
\[ A \mathbf{x} = \lambda_i \mathbf{x} \]
Step 3: Formulate Augmented Matrix
Formulate the augmented matrix by subtracting $\lambda_i \mathbf{I}$ from matrix $A$, where $\mathbf{I}$ is the identity matrix.
\[ (A – \lambda_i \mathbf{I}) \mathbf{x} = \mathbf{0} \]
Step 4: Solve the Homogeneous System
Solve the homogeneous system of linear equations $(A – \lambda_i \mathbf{I}) \mathbf{x} = \mathbf{0}$ to find the null space of the matrix.
Step 5: Normalize Eigenvectors
Normalize the eigenvectors obtained in the previous step by dividing each vector by its norm.
This step ensures that the eigenvectors have a unit length.
For a given eigenvector $\mathbf{v}$, its normalized version $\mathbf{v}_{\text{norm}}$ can be obtained as follows:
\[ \mathbf{v}_{\text{norm}} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \]
where $\|\mathbf{v}\|$ represents the Euclidean norm (magnitude) of the vector $\mathbf{v}$.
