Eigenvalues and Eigenvectors
Definition and Significance
- An eigenvector of a square matrix $A$ is a non-zero vector $\mathbf{v}$ that, when multiplied by $A$, results in a scalar multiple of itself.
- This scalar is called the eigenvalue corresponding to that eigenvector.
Mathematically, this is expressed as:
$$A\mathbf{v} = \lambda\mathbf{v}$$
where $\lambda$ is the eigenvalue.
NoteEigenvalues and eigenvectors provide crucial information about the behavior of linear transformations represented by matrices.
Characteristic Polynomial
For a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the characteristic polynomial is given by:
$$p(\lambda) = \det(A - \lambda I) = \lambda^2 - (a+d)\lambda + (ad-bc)$$
where $I$ is the 2x2 identity matrix.
The roots of this polynomial are the eigenvalues of the matrix. For a 2x2 matrix, we can find these roots using the quadratic formula:
$$\lambda = \frac{(a+d) \pm \sqrt{(a+d)^2 - 4(ad-bc)}}{2}$$
ExampleConsider the matrix \( A \): \[ A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \] The characteristic polynomial is given by: \[ p(\lambda) = \det(A - \lambda I) = \begin{vmatrix} 3 - \lambda& 1 \\ 1 & 3 - \lambda\end{vmatrix} \] Expanding the determinant: \[ (3 - \lambda)(3 - \lambda) - (1)(1) = \lambda^2 - 6\lambda + 8 \] Solving the quadratic equation: \[\lambda^2 - 6\lambda + 8 = 0 \] Factoring: \[ (\lambda - 4)(\lambda - 2) = 0 \] Thus, the eigenvalues are: \[\lambda_1 = 4, \quad\lambda_2 = 2 \]
Finding Eigenvectors
Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the equation $(A - \lambda I)\mathbf{v} = \mathbf{0}$ for each eigenvalue.
ExampleBased on the previous example:
For the eigenvector corresponding to \(\lambda_1 = 4 \): \[ (A - 4I) \mathbf{v} = 0 \]\[\begin{bmatrix} 3 - 4 & 1 \\ 1 & 3 - 4 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]\[\begin{bmatrix} -1 & 1 \\ 1 & -1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \] From the first equation: \[ - v_1 + v_2 = 0 \Rightarrow v_1 = v_2 \] Thus, an eigenvector for \(\lambda_1 = 4 \) is: \[\mathbf{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix} \] Similarly, for \(\lambda_2 = 2 \): \[ (A - 2I) \mathbf{v} = 0 \]\[\begin{bmatrix} 3 - 2 & 1 \\ 1 & 3 - 2 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \]\[\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} \] From the first equation: \[ v_1 + v_2 = 0 \Rightarrow v_1 = -v_2 \] Thus, an eigenvector for \(\lambda_2 = 2 \) is: \[\mathbf{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix} \]
Diagonalization of 2x2 Matrices
- Diagonalization is a process of transforming a matrix into a diagonal matrix.
- For 2x2 matrices with distinct real eigenvalues, this process is always possible.
Process of Diagonalization
- Find the eigenvalues of the matrix.
- Find the corresponding eigenvectors.
- Form the matrix $P$ with eigenvectors as columns.
- Form the diagonal matrix $D$ with eigenvalues on the diagonal.
The diagonalization is then given by:
$$A = PDP^{-1}$$
NoteDiagonalization is extremely useful as it simplifies many matrix operations, especially when dealing with matrix powers.
ExampleFor the matrix \( A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \): \[ P = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \quad D = \begin{bmatrix} 4 & 0 \\ 0 & 2 \end{bmatrix} \] We can verify that: \[ A = P D P^{-1} \]
Applications to Powers of 2x2 Matrices
- One of the most powerful applications of diagonalization is in calculating high powers of matrices efficiently. The formula for this is: $$A^n = PD^nP^{-1}$$
- This is particularly useful because $D^n$ is simply a diagonal matrix with the eigenvalues raised to the $n$th power.
When dealing with large powers of matrices, always consider diagonalization as a potential method to simplify calculations.
ExampleTo find \( A^{10} \) for \( A = \begin{bmatrix} 3 & 1 \\ 1 & 3 \end{bmatrix} \): \[ A^{10} = P \begin{bmatrix} 4^{10} & 0 \\ 0 & 2^{10} \end{bmatrix} P^{-1} \] This is much easier to calculate than multiplying \( A \) by itself 10 times!
Real-World Applications
Population Movement
- Eigenvalues and eigenvectors can model population movement between two towns.
- The eigenvector corresponding to the largest eigenvalue represents the long-term distribution of population between the towns.
If \( A = \begin{bmatrix} 0.7 & 0.3 \\ 0.2 & 0.8 \end{bmatrix} \) represents the yearly movement probabilities between two towns, the eigenvector \(\begin{bmatrix} 0.6 \\ 0.4 \end{bmatrix} \) corresponding to the largest eigenvalue \( 1 \) suggests that in the long term, 60% of the population will be in town 1 and 40% in town 2.
Predator-Prey Models
- In ecology, eigenvalues can help analyze the stability of predator-prey systems.
- The nature of the eigenvalues (real or complex, positive or negative) provides information about the system's behavior over time.
Students often forget that while eigenvalues give information about the long-term behavior, the eigenvectors are crucial for understanding the specific nature of that behavior.
NoteThis connection highlights the interdisciplinary nature of mathematics and its applications in various fields of science and engineering.
TipWhen studying eigenvalues and eigenvectors, always try to visualize their geometric interpretation. This can greatly enhance understanding of their mathematical and practical significance.
