Eigen Decomposition of a Matrix

Eigen decomposition is a method used in linear algebra to break down a square matrix into simpler components called eigenvalues and eigenvectors. This process helps us understand how a matrix behaves and how it transforms data.

For Example - Eigen decomposition is particularly useful in fields like Physics, Machine learning, and Computer graphics, as it simplifies complex calculations.

In this article, we will cover the fundamentals of Eigen decomposition, its significance, and its practical uses in different mathematical and real-world scenarios.

Fundamental Theory of Eigen Decomposition

Eigen decomposition separates a matrix into its eigenvalues and eigenvectors. Mathematically, for a square matrix A, if there exists a scalar λ (eigenvalue) and a non-zero vector v (eigenvector) such that:

Av = λv

Where:

• A is the matrix.
• λ is the eigenvalue.
• v is the eigenvector.

Then, the matrix A can then be represented as:

A=VΛV^-1

Where:

• V is the matrix of eigenvectors.
• Λ is the diagonal matrix of eigenvalues.
• V^-1 is the inverse of the matrix.

This decomposition is significant because it transforms matrix operations into simpler, scalar operations involving eigenvalues, making computations easier.

How to Perform Eigen decomposition?

To perform Eigen decomposition on a matrix, follow these steps:

• Step 1: Find the Eigenvalues:

Solve the characteristic equation:

det (A−λI=0

Here, A is the square matrix, λ is the eigenvalue, and I is the identity matrix of the same dimension as A.

• Step 2: Find the Eigenvectors:

For each eigenvalue λ, substitute it back into the equation:

(A−λI)v=0

This represents a system of linear equations where v is the eigenvector corresponding to the eigenvalue λ.

• Step 3: Construct the Eigenvector Matrix V:

Place all the eigenvectors as columns in the matrix V. If there are n distinct eigenvalues, V will be an n×n matrix..

• Step 4 Form the Diagonal Matrix Λ:

Construct a diagonal matrix Λ by placing the eigenvalues on its diagonal:

• Step 5: Calculate the Inverse of V:

Find V-1, the inverse of the eigenvector matrix V, if the matrix is invertible.

Example of Eigen Decomposition

Define the matrix, A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}

Find the eigenvalues by solving det(A - λI) = 0

The characteristic equation is: A−λI = 0

\Rightarrow \begin{vmatrix} 4 - \lambda & 1 \\ 2 & 3 - \lambda \end{vmatrix} = 0

\Rightarrow (4 - \lambda)(3 - \lambda) - (2)(1) = 0

\Rightarrow \lambda^2 - 7\lambda + 10 = 0

\Rightarrow \lambda_1 = 5, and \lambda_2 = 2

Find the eigenvectors corresponding to each eigenvalue

For, \lambda_1 = 5, solve: (A - 5I)v = 0:

\begin{bmatrix} -1 & 1 \\ 2 & -2 \end{bmatrix}

\Rightarrow \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = 0

\Rightarrow v_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}

For, \lambda_2 = 2, solve: (A - 2I)v = 0:

\begin{vmatrix} 2 & 1 \\ 2 & 1 \end{vmatrix} = 0

\Rightarrow \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = 0

\Rightarrow v_2 = \begin{bmatrix} -\frac{1}{2} \\ 1 \end{bmatrix}

Form the matrix V of eigenvectors and Λ of eigenvalues

V = \begin{bmatrix} 1 & -\frac{1}{2} \\ 1 & 1 \end{bmatrix}, \quad

\Lambda = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}

Perform the Eigen decomposition

A = V \Lambda V^{-1}

Optionally, compute the inverse of V

V^{-1} = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ \\ -\frac{2}{3} & \frac{2}{3} \end{bmatrix}

Importance of Eigen decomposition

Eigen decomposition is widely used because it makes complex tasks simpler:

• Simplifying Matrix Powers: It helps in easily calculating powers of matrices, which is useful in solving equations and modeling systems.

• Data Simplification: It is used in techniques like PCA to reduce large datasets into fewer dimensions, making them easier to analyze.

• Physics: In quantum mechanics, it helps in understanding how systems change over time.

• Image Processing: It is used in tasks like image compression and enhancement, making handling images more efficient.