# Approximate update methods for the singular value decomposition

> Source: https://ollybritton.com/misc/publications/approximate-svd-updates/ · Updated: 2026-10-11 · Tags: uni

During [Part C](https://ollybritton.com/notes/uni/part-c/) I worked with [Prof. Yuji Nakatsukasa](https://people.maths.ox.ac.uk/nakatsukasa/) for my dissertation on approximate update methods for the singular value decomposition. I looked at the following problem: if you have a matrix $A \in \mathbb R^{n \times n}$ whose singular value decomposition $A = U \Sigma V^\top$ is already known, how can you leverage this information to approximate the singular values of a perturbation $A + E$, where $E$ is small?

![The perturbed matrix A + E becomes nearly diagonal when expressed in the original singular-vector bases.](britton-2026-dissertation-central-identity.png)

You can [download my dissertation here](britton-2026-masters-dissertation.pdf).

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