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Tutorial: Sherman Morrison Formula

Revision en7, by McDic, 2020-02-22 10:48:12

### Introduction

Hello. I would like to share some nice stuff, called Sherman-Morrison formula. If you know this formula, it's easier to understand how BFGS update works in convex optimization and machine learning. But in this topic, we will just simply focus on this formula and learn how to calculate inverse matrix of rank-$1$ updated invertible matrix faster.

The motivation of this post is that my problem using Sherman Morrison formula is rejected by multiple coordinators.

### Definition

You have invertible $\mathbb{R}^{n \times n}$ matrix $A$, and $\mathbb{R}^{n}$ vectors $u$ and $v$. Then $(A + uv^{T})^{-1} = A^{-1} - \frac{A^{-1}uv^{T}A^{-1}}{v^{T}A^{-1}u + 1}$. If $v^{T}A^{-1}u = -1$, then $A + uv^{T}$ is non-invertible.

### Time complexity

You can calculate Sherman Morrison formula in $O(n^2)$, because $A^{-1}u$, $v^{T}A^{-1}$ and $v^{T}A^{-1}$ are vector.

Normally you need $O(n^3)$ time complexity to calculate inverse matrix except for some special matrix(diagonal matrix etc). But, since you can calculate Sherman-Morrison formula in $O(n^2)$, if you find some matrix can be represented as rank-$1$ updated matrix of special matrix, you can calculate inverse of that matrix in $O(n^2)$.

### Generalization

There is a generalization of this formula called Woodbury Matrix Identity.

### Sample problem

This problem is that rejected problem.

You have matrix $A = \begin{bmatrix} 1 & 2 & \cdots & n \\ n+1 & n+2 & \cdots & 2n \\ \cdots & \cdots & \cdots & \cdots \\ n^{2}-n+1 & n^{2}-n+2 & \cdots & n^{2} \end{bmatrix} + diag(c_{1}, c_{2}, \ldots, c_{n})$.

In $i$-th query, you will select any row or column of $A$ and multiply this by $x_{i}$. Compute $A^{-1}$ if $A$ is invertible after each query.

Constraints: $2 \le n \le 200$, $1 \le q \le 200$, $0 \lt c_{i}$, $x_{i} \neq 0$. #### History

Revisions Rev. Lang. By When Δ Comment
en10 McDic 2020-02-22 16:42:59 10
en9 McDic 2020-02-22 12:15:12 14
en8 McDic 2020-02-22 12:14:58 6
en7 McDic 2020-02-22 10:48:12 68
en6 McDic 2020-02-22 10:45:43 21
en5 McDic 2020-02-22 10:43:42 16
en4 McDic 2020-02-22 10:43:04 158
en3 McDic 2020-02-22 10:39:15 42
en2 McDic 2020-02-22 10:38:16 105 (published)
en1 McDic 2020-02-22 10:31:33 1826 Initial revision (saved to drafts)