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Understanding MATLABs Left (`\`) and Right (`/`) Division Operators

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In MATLAB, matrix division operators are used to solve systems of linear equations efficiently, with the two main operators being the left division operator (\) and the right division operator (/). Their behavior depends on the orientation of the matrices involved and the type of problem you want to solve. Here's an explanation of Left Division (\) versus Right Division (/):

1. Left Division (\) — Backslash Operator

x = A \ B;

Purpose:
Solve the system of linear equations:

[ A x = B ]

for x.

Usage scenarios:

  • When A is a matrix and B is a vector or matrix, this computes the solution x that satisfies the equation.
  • It performs matrix left division, essentially computing:

[ x = A^{-1} B ]

(But more efficiently and robustly than explicitly calculating the inverse.)

Example:

A = [1, 2; 3, 4];
B = [5; 6];
x = A \ B;  % solves for x in A*x = B

2. Right Division (/) — Slash Operator

x = B / A;

Purpose:
Solve the system of linear equations:

[ x A = B ]

for x.

Usage scenarios:

  • When you want x such that x A = B.
  • Equivalent to:

[ x = B A^{-1} ]

(Again, MATLAB uses optimized routines rather than explicitly computing the inverse.)

Example:

A = [1, 2; 3, 4];
B = [5; 6];
x = B / A;  % solves for x in x*A = B

Key differences summarized

| Operator | Equation solved | How MATLAB interprets | Typical application | |-------------------|----------------------------------------|-----------------------------------------------------|---------------------------------------------| | Left division (\) | A \ B | solves (A x = B) | Solving systems (A x = B) directly | | Right division (/) | B / A | solves (x A = B) | Solving systems (x A = B) or in data rearrangements |

Note:

  • Both operators do not simply invert matrices; they use specialized algorithms like LU, QR, or SVD decomposition, which are more preferred for numerical stability.
  • For square, full-rank matrices**, A \ B is equivalent to inv(A)*B but computed more efficiently and accurately.
  • For non-square or rank-deficient matrices, MATLAB computes least-squares solutions using \.

Summary

| Operation | Use case | Example syntax | |------------------------------|-------------------------------------------------------|---------------------------------------------| | Solve (A x = B) | Left division: A \ B | x = A \ B; | | Solve (x A = B) | Right division: B / A | x = B / A; |

Let me know if you'd like specific examples or further clarification!