What are Eigen Value and Eigen Function. Define the Degeneracy of Eigen Value.
EigenVectors are non-zero vectors that remain in the same direction (up to scaling) after a matrix transformation. In other words, when we apply a square matrix to an eigenvector, the resulting vector is simply a scaled version of the original vector, but it still points in the same direction. This scaling factor is the corresponding eigenvalue associated with that eigenvector.
Mathematically, for a square matrix \(\mathbf{A}\) and an eigenvector \(\mathbf{v}\) with its corresponding eigenvalue \(\lambda\), the relationship is given by:
\[ \mathbf{A} \cdot \mathbf{v} = \lambda \cdot \mathbf{v} \]
where \(\cdot\) represents matrix multiplication.
Degeneracy of Eigenvalues :
The concept of the degeneracy of eigenvalues arises when a square matrix has repeated eigenvalues. In such cases, we say that the eigenvalues are degenerate. The degeneracy of an eigenvalue refers to the number of linearly independent eigenvectors associated with that particular eigenvalue.
Here are some examples to elaborate all the above concepts :
Example 1 :
Consider the following 2x2 matrix \(\mathbf{A}\):
\[ \mathbf{A} = \begin{bmatrix} 3 & 1 \\ 1 & 2 \end{bmatrix} \]
To find the eigenvalues and eigenvectors, we need to solve the characteristic equation:
\[ \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \]
where \(\lambda\) is the eigenvalue and \(\mathbf{I}\) is the identity matrix.
By solving the characteristic equation, we find two eigenvalues: \(\lambda_1 = 2\) and \(\lambda_2 = 3\).
For \(\lambda_1 = 2\), the corresponding eigenvector \(\mathbf{v}_1\) is \(\begin{bmatrix} 1 \\ -1 \end{bmatrix}\).
For \(\lambda_2 = 3\), the corresponding eigenvector \(\mathbf{v}_2\) is \(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\).
Example 2 :
Consider the following 3x3 matrix \(\mathbf{B}\):
\[ \mathbf{B} = \begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix} \]
By solving the characteristic equation, we find three eigenvalues: \(\lambda_1 = 2\), \(\lambda_2 = 2\), and \(\lambda_3 = 3\).
For \(\lambda_1 = 2\), there are two linearly independent eigenvectors: \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix}\) and \(\mathbf{v}_2 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}\).
For \(\lambda_3 = 3\), there is one eigenvector: \(\mathbf{v}_3 = \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}\).
In this example, \(\lambda_1 = 2\) has a degeneracy of 2, while \(\lambda_3 = 3\) has a degeneracy of 1.
Summary :
- Eigenvalues and eigenvectors are special properties of square matrices.
- Eigenvectors remain in the same direction (up to scaling) after a matrix transformation, and the scaling factor is the eigenvalue.
- The degeneracy of eigenvalues refers to the number of linearly independent eigenvectors associated with a particular eigenvalue.
- Higher degeneracy provides more freedom in the diagonalization process of the matrix.
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