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Using SVD to Repair a Non-Positive-Definite Matrix for Cholesky

Article Quant Q&A · Author: atastix
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Summary

The note gives a practical use of singular value decomposition in quantitative finance: diagnosing a square matrix that cannot be factorized by Cholesky because it is not positive definite. SVD can help assess whether the matrix can be adjusted to become positive definite while remaining close to the original, so Cholesky can then be applied.

The example is a brief conceptual answer, not a worked procedure. It provides no specific adjustment method, quantitative evidence, or guidance on how to measure whether the revised matrix is materially different. It also does not explain how to interpret SVD components for stock price behavior, which was the original question. The technique is therefore relevant to matrix preparation, but the note offers limited detail for broader SVD analysis or trading use.

Key ideas

  • SVD can be used when a square matrix fails the positive-definiteness requirement for Cholesky factorization.
  • The decomposition can help evaluate whether a nearby positive-definite matrix can be constructed.
  • The note gives no specific repair algorithm or criterion for acceptable deviation from the original matrix.

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# Resources to learn the applications of SVD in quant finance?


# Resources to learn the applications of SVD in quant finance?












I have been searching for quite a while on how singular value decomposition is used in analyzing stock price behavior. I know how to perform it on a matrix of stock prices, have the results in python but have no idea on how to interpret the results. Tried searching all of youtube and the net and cant find any explanations.

Note that I am not looking for an explanation of the SVD concept, but how its interpreted for its finance applications.

## Answer by Dimitri Vulis (score 3)

https://quant.stackexchange.com/a/60929

An example of typical use of SVD:

Suppose you have a square matrix. You would like to apply Cholesky decomposition. But Cholesky complains that the matrix is not positive definite. So you call SVD instead.

You can then see whether you can tweak the matrix to obtain a positive definite matrix not materially different from your original matrix, and use Cholesky.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.