Introduction
- Eigenvalues and eigenvectors are one of the concepts in matrices in linear algebra.
- Eigenvectors are characteristic non-zero vectors of matrices.
- The factor ‘λ’ of an eigenvector is called the eigenvalue.
- Eigenvectors remain in the same direction when a linear transformation is applied.
The transformation equation
\[\LARGE
Ax = \lambda x
\]
Trivial and non-trivial solutions
- When the vector x is all zeros, then it is known as a trivial solution.
- When the vector x is zero, it is known as a non-trivial solution.
\[\Large
x =
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix}
~~~~~\color{purple}{Trivial~solution}
\]
\[\Large x = \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix} ~~~~~\color{purple}{Non-trivial~solution} \]
\[\Large x = \begin{bmatrix} 3 \\ 2 \\ -1 \end{bmatrix} ~~~~~\color{purple}{Non-trivial~solution} \]
- Eigenvectors must be non-zero, that means they need to be non-trivial solutions.
An eigenvector itself can entirely be zero.