- Hands-On Mathematics for Deep Learning
- Jay Dawani
- 336字
- 2021-06-18 18:55:10
Permutations
In the example on solving systems of linear equations, we swapped the positions of rows 2 and 3. This is known as a permutation.
When we are doing triangular factorization, we want our pivot values to be along the diagonal of the matrix, but this won't happen every time—in fact, it usually won't. So, instead, what we do is swap the rows so that we get our pivot values where we want them.
But that is not their only use case. We can also use them to scale individual rows by a scalar value or add rows to or subtract rows from other rows.
Let's start with some of the more basic permutation matrices that we obtain by swapping the rows of the identity matrix. In general, we have n! possible permutation matrices that can be formed from an nxn identity matrix. In this example, we will use a 3×3 matrix and therefore have six permutation matrices, and they are as follows:
This matrix makes no change to the matrix it is applied on.
This matrix swaps rows two and three of the matrix it is applied on.
This matrix swaps rows one and two of the matrix it is applied on.
This matrix shifts rows two and three up one and moves row one to the position of row three of the matrix it is applied on.
This matrix shifts rows one and two down one and moves row three to the row-one position of the matrix it is applied on.
This matrix swaps rows one and three of the matrix it is applied on.
It is important to note that there is a particularly fascinating property of permutation matrices that states that if we have a matrix and it is invertible, then there exists a permutation matrix that when applied to A will give us the LU factor of A. We can express this like so:

- 數(shù)據(jù)庫技術(shù)與應(yīng)用教程(Access)
- 復雜性思考:復雜性科學和計算模型(原書第2版)
- Modern Programming: Object Oriented Programming and Best Practices
- 大數(shù)據(jù)可視化
- MySQL從入門到精通(第3版)
- Hadoop大數(shù)據(jù)實戰(zhàn)權(quán)威指南(第2版)
- 大數(shù)據(jù)治理與安全:從理論到開源實踐
- 數(shù)據(jù)科學實戰(zhàn)指南
- 數(shù)據(jù)分析師養(yǎng)成寶典
- Hands-On System Programming with C++
- 大數(shù)據(jù)分析:R基礎(chǔ)及應(yīng)用
- 數(shù)據(jù)分析思維:產(chǎn)品經(jīng)理的成長筆記
- Kubernetes快速進階與實戰(zhàn)
- 碼上行動:利用Python與ChatGPT高效搞定Excel數(shù)據(jù)分析
- 精通Neo4j