Matrices represent data and transformations, but their structure can sometimes be complex. Matrix decomposition techniques offer ways to factorize a matrix into simpler, constituent matrices. This factorization often makes it easier to understand the matrix's properties and perform certain calculations efficiently.

This chapter introduces key matrix decomposition methods used frequently in machine learning contexts. You will learn about:

• Singular Value Decomposition (SVD): A powerful decomposition ($A = U\Sigma V^T$) applicable to any $m \times n$ matrix. We will look at its components (singular values and vectors), its geometric interpretation, and its use in dimensionality reduction, noise reduction, and data compression.
• LU Decomposition: Factoring a square matrix into a product of a lower triangular matrix ($L$) and an upper triangular matrix ($U$), often written as $A = LU$. This is particularly useful for solving systems of linear equations $Ax=b$.
• QR Decomposition: Decomposing a matrix into an orthogonal matrix ($Q$) and an upper triangular matrix ($R$), such that $A = QR$. This method is common in algorithms related to least-squares problems.

We will discuss the relationship between SVD and the eigen-decomposition covered in the previous chapter. Practical implementation of these decompositions using Python libraries like NumPy and SciPy will also be covered, allowing you to apply these techniques to data.