Matrix decomposition is a crucial technique in linear algebra, breaking down matrices into simpler components to simplify computations and reveal structural properties. Common types include:
1. Singular Value Decomposition (SVD)
A powerful method for factorizing any matrix into three components:
- U: Orthogonal matrix of left singular vectors
- Σ: Diagonal matrix of singular values
- V: Orthogonal matrix of right singular vectors
2. LU Decomposition
Splits a matrix into a lower triangular matrix (L) and an upper triangular matrix (U).
- Purpose: Simplifies solving linear systems (e.g., $Ax = b$)
- Formula: $A = LU$
3. QR Decomposition
Decomposes a matrix into an orthogonal matrix (Q) and a triangular matrix (R).
- Application: Used in least squares problems and numerical stability
- Formula: $A = QR$
4. Cholesky Decomposition
Special case for symmetric positive-definite matrices:
- Result: $A = LL^T$ where L is lower triangular
- Use Case: Efficient for solving systems in statistics and engineering
For deeper exploration, check our Matrix Multiplication tutorial to understand the foundational operations. 📚✨