Rewriting a Matrix Using Eigenvalues and Eigenvectors: A Guide to Eigendecomposition

One of the most elegant and insightful ways to understand a matrix is through its eigendecomposition. This approach allows us to express a square matrix in terms of its eigenvalues and eigenvectors, revealing how the matrix stretches or compresses space along certain directions. In this article, we will explore how and when a matrix can be rewritten using its eigenvalues and eigenvectors.

The Eigendecomposition Formula

If a matrix \( A \in \mathbb{R}^{n \times n} \) is diagonalizable, it can be expressed as:

\[ A = P D P^{-1} \]

• \( A \) is the original matrix
• \( P \) is a matrix whose columns are the eigenvectors of \( A \)
• \( D \) is a diagonal matrix containing the corresponding eigenvalues
• \( P^{-1} \) is the inverse of \( P \)

When is a Matrix Diagonalizable?

Not every matrix can be decomposed this way. A matrix is diagonalizable if and only if it has enough linearly independent eigenvectors to form the matrix \( P \). A few common cases are:

• All real symmetric matrices are always diagonalizable.
• Non-symmetric matrices may or may not be diagonalizable depending on the algebraic and geometric multiplicity of their eigenvalues.

Worked Example

Let’s consider the matrix:

\[ A = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix} \]

Suppose this matrix has the following eigenvalues and eigenvectors:

• Eigenvalues: \( \lambda_1 = 5, \lambda_2 = 2 \)
• Eigenvectors: \[ \vec{v}_1 = \begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \vec{v}_2 = \begin{bmatrix} 1 \\ -2 \end{bmatrix} \]

Then we form the matrices:

\[ P = \begin{bmatrix} 1 & 1 \\ 1 & -2 \end{bmatrix}, \quad D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix} \]

So the eigendecomposition becomes:

\[ A = P D P^{-1} \]

To verify this, you would compute \( P^{-1} \), multiply \( PDP^{-1} \), and confirm that it gives back the original matrix \( A \).

Why Eigendecomposition Matters

Rewriting a matrix in this form has several powerful applications in mathematics and applied fields:

Application	How It Helps
Matrix Powers	Easily compute \( A^n = P D^n P^{-1} \)
Solving Differential Equations	Diagonalization simplifies linear systems
Principal Component Analysis (PCA)	PCA uses eigenvectors of the covariance matrix to find principal directions
Quantum Mechanics	Eigenvalues correspond to measurable quantities like energy levels

Conclusion

Yes, a matrix can be rewritten using its eigenvalues and eigenvectors—through eigendecomposition—if it is diagonalizable. This rewriting allows for more efficient computation, deeper insight into the transformation properties of the matrix, and serves as the backbone for many advanced applications in both pure and applied mathematics.