Last modified: December 22, 2024
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Eigenvalue Decomposition (EVD)
Eigenvalue Decomposition (EVD), also known as Eigendecomposition, is a fundamental operation in linear algebra that breaks down a square matrix into a simpler form defined by its eigenvalues and eigenvectors. This decomposition provides deep insights into the properties and structure of a matrix, enabling simplifications in numerous computations such as matrix powers, transformations, and the solution of systems of linear equations. In many applications — from vibration analysis in engineering to principal component analysis in data science — EVD plays a critical role.
The key idea is that certain square matrices can be represented as a product of a matrix of their eigenvectors and a diagonal matrix of their eigenvalues. This diagonalization separates the scaling factors (eigenvalues) from the directions of transformations (eigenvectors) encoded by the original matrix.
Mathematical Formulation
Consider an $n \times n$ matrix $A$. The Eigenvalue Decomposition of $A$ is given by:
$$A = P D P^{-1},$$ where:
I. Eigenvalues ($\lambda_i$):
The eigenvalues of $A$ are the roots of the characteristic polynomial:
$$\det(A - \lambda I) = 0.$$
If we solve this polynomial equation and find $n$ eigenvalues (not necessarily distinct), we denote them as $\lambda_1, \lambda_2, \ldots, \lambda_n$.
II. Eigenvectors ($v_i$):
For each eigenvalue $\lambda_i$, we find the corresponding eigenvector $v_i$ by solving:
$$(A - \lambda_i I) v_i = 0.$$
Each eigenvector $v_i$ is a nonzero vector associated with the eigenvalue $\lambda_i$.
III. Eigenvector Matrix ($P$) and Eigenvalue Matrix ($D$):
Once we have the set of eigenvalues and eigenvectors:
Construct $D$ as a diagonal matrix whose diagonal entries are the eigenvalues:
$$D = \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 & \lambda_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n \end{bmatrix}$$
Construct $P$ such that its columns are the eigenvectors $v_1, v_2, \ldots, v_n$:
$$P = \begin{bmatrix} | & | & & | \\ v_1 & v_2 & \cdots & v_n \\ | & | & & | \end{bmatrix}$$
If $A$ is diagonalizable, and if the $v_i$ are chosen to be linearly independent, then $P$ is invertible and $A = P D P^{-1}$.
Derivation
The derivation of EVD follows naturally from the definition of eigenvalues and eigenvectors. For each eigenvalue $\lambda_i$, we have:
$$A v_i = \lambda_i v_i.$$
If we gather all eigenvectors into the matrix $P$ and consider how $A$ acts on the columns of $P$:
$$A [v_1 \, v_2 \, \cdots \, v_n] = [A v_1 \, A v_2 \, \cdots \, A v_n] = [\lambda_1 v_1 \, \lambda_2 v_2 \, \cdots \, \lambda_n v_n] = P D.$$
If $P$ is invertible, we can write:
$$A = P D P^{-1}.$$
Not every matrix is guaranteed to have a full set of $n$ linearly independent eigenvectors. If it does, the matrix is said to be diagonalizable, and the above decomposition is possible. If not, the matrix cannot be decomposed purely into this form.
Algorithm Steps
I. Find Eigenvalues:
- Form the characteristic polynomial $\det(A - \lambda I) = 0$.
- Solve for $\lambda$. This may be done analytically for small matrices or numerically for larger matrices.
II. Find Eigenvectors:
- For each eigenvalue $\lambda_i$, solve $(A - \lambda_i I)v_i = 0$.
- Ensure each eigenvector $v_i$ is normalized or scaled consistently.
III. Form $P$ and $D$:
- Construct $D$ as the diagonal matrix with eigenvalues on the diagonal.
- Construct $P$ with eigenvectors as columns.
IV. Verify Invertibility of $P$:
- If $P$ is invertible, then $A = P D P^{-1}$.
Example
Let:
$$A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}$$
I. Find Eigenvalues:
The characteristic polynomial:
$$\det(A - \lambda I) = \det\begin{bmatrix}4 - \lambda & 1 \ 2 & 3 - \lambda\end{bmatrix} = (4-\lambda)(3-\lambda) - 2 = 0.$$
Expanding:
$$(4-\lambda)(3-\lambda) - 2 = (12 -7\lambda + \lambda^2) - 2 = \lambda^2 -7\lambda +10=0.$$
Solve $\lambda^2 -7\lambda +10=0$:
The roots are $\lambda_1=5$ and $\lambda_2=2$.
II. Find Eigenvectors:
For $\lambda_1 = 5$:
$$(A - 5I)=\begin{bmatrix}-1 & 1 \ 2 & -2\end{bmatrix}.$$
Solve $(A-5I)v=0$ leads to $v_1 = [1,1]^T$.
For $\lambda_2 = 2$:
$$(A - 2I)=\begin{bmatrix}2 & 1 \ 2 & 1\end{bmatrix}$$
Solve $(A-2I)v=0$ leads to $v_2 = [-1,2]^T$.
III. Form $P$ and $D$:
$$P = \begin{bmatrix}1 & -1 \ 1 & 2\end{bmatrix}, \quad D = \begin{bmatrix}5 & 0 \ 0 & 2\end{bmatrix}$$
Thus:
$$A = P D P^{-1}.$$
Advantages
I. Simplification of Computations:
Once in the form $A = P D P^{-1}$, computing powers of $A$ or applying certain transformations becomes much simpler. For example, $A^k = P D^k P^{-1}$, and since $D$ is diagonal, raising it to a power is straightforward.
II. Insights into Matrix Structure:
The eigendecomposition reveals the intrinsic "modes" of the linear transformation represented by $A$. Eigenvalues show how the transformation scales each eigen-direction, and eigenvectors show the directions of these fundamental modes.
III. Numerical Stability in Some Computations:
Working with $D$ instead of $A$ can improve numerical stability and make some algorithms more efficient, particularly in areas like principal component analysis, spectral clustering, and other advanced data analysis tasks.
Limitations
I. Not All Matrices Are Diagonalizable:
Some matrices cannot be broken down into a pure eigen decomposition if they do not have enough linearly independent eigenvectors. For such matrices, more generalized decompositions like the Jordan normal form are required.
II. Computational Cost for Large Matrices:
Finding eigenvalues and eigenvectors for large matrices can be computationally expensive. Efficient numerical algorithms and approximations exist, but they may still be costly for very large systems.
III. Complex Eigenvalues:
For real matrices, eigenvalues can be complex. While this is not a fundamental limitation, it means we must consider complex arithmetic when performing the decomposition, which may not be desired in some real-world applications.