How to Calculate the SVD from Scratch with Python

Last Updated on October 18, 2019

Matrix decomposition, also known as matrix factorization, involves describing a given matrix using its constituent elements.

Perhaps the most known and widely used matrix decomposition method is the Singular-Value Decomposition, or SVD. All matrices have an SVD, which makes it more stable than other methods, such as the eigendecomposition. As such, it is often used in a wide array of applications including compressing, denoising, and data reduction.

In this tutorial, you will discover the Singular-Value Decomposition method for decomposing a matrix into its constituent elements.

After completing this tutorial, you will know:

• What Singular-value decomposition is and what is involved.
• How to calculate an SVD and reconstruct a rectangular and square matrix from SVD elements.
• How to calculate the pseudoinverse and perform dimensionality reduction using the SVD.

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Let’s get started.

• Update Mar/2018: Fixed typo in reconstruction. Changed V in code to VT for clarity. Fixed typo in the pseudoinverse equation.
• Update Apr/2019: Fixed a small typo re array sizes in the explanation of the SVD
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