To actively decompose a given matrix, Singular Value Decomposition (SVD) utilizes three matrices. The SVD technique is widely used in **machine learning** for dimensionality reduction. By utilizing the decomposed matrices, we can actively approximate the original matrix with a **lower-rank **representation. The process involves passively decomposing the original matrix into three matrices using SVD. Then actively using them to obtain the lower-rank approximation.

Given an **m x n** matrix A, SVD factorizes **A **into three matrices:

**A = U * Σ * V^T**

where **U** is an **m x r** matrix whose columns are the left singular vectors of **A**, and **Σ **is an **r x r **diagonal matrix containing the singular values of **A**. **V **is an **n x r** matrix whose columns are the right singular vectors of **A**.

To obtain a lower-rank approximation of matrix A, we can use the singular values in **Σ**. It is arranged in decreasing order. By selecting the first k singular values and their corresponding **left **and **right **singular vectors. We can obtain a set of matrices: the first **k **columns of **U**, the first k rows of **Σ**, and the first k rows of **V^T**. These matrices are multiplying together using matrix multiplication to obtain a new matrix **B **with dimensions **k x n**.

**B = Uk * Σk * Vk^T**

In Singular Value Decomposition (SVD), we can use it to reduce dimensionality by selecting a subset of the largest singular values and their corresponding singular vectors. This will allow us to approximate the original matrix with a lower-rank representation. The process involves actively selecting the subset of singular values and singular vectors. Then using the obtain the lower-rank approximation of the original matrix. This can be useful for reducing the computational complexity of a dataset and identifying important features.

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