What is the difference between backward slash vs forward slash in MATLAB?

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I have a failry simple question in MATLAB. What s the difference between the backslash operator vs the forward slash operator. For example,
A=[4,12;6,8];
b=[6,12;14,8];
x1 = A/b;
1.1333 -0.2000
0.5333 0.2000
%which is different from:
x2 = A\b;
3.0000 0
-0.5000 1.0000
I am asking because I am trying to convert a simple line of code from MATLAB to c++ which it turns out there's no forward slash in c++ unfortunately.

Accepted Answer

William Rose
William Rose on 20 Jun 2022
X=A\B computes X=inv(A)*B.
Y=A/B computes Y=A*inv(B)
A=rand(2,2); B=rand(2,2);
X1=A\B
X1 = 2×2
8.5138 7.1476 -3.5676 -2.2030
X2=inv(A)*B
X2 = 2×2
8.5138 7.1476 -3.5676 -2.2030
Y1=A/B
Y1 = 2×2
0.2325 -0.2566 -0.0595 0.7033
Y2=A*inv(B)
Y2 = 2×2
0.2325 -0.2566 -0.0595 0.7033
When I look at A\B, I try to remember that the A looks like it is "under" the divide sign, which reminds me that A is the denominator in A\B. And it comes first, so inv(A) is before B in the (non-commutative) multiplication.
  7 Comments
Stephen23
Stephen23 on 20 Jun 2022
Edited: Stephen23 on 20 Jun 2022
" Matlab's left divide may not use the equation I gave above - @John D'Errico says it doesn't, and I trust him."
Even better is to read the MLDIVIDE() documentation yourself:
"The equation I gave in my comment (not my original answer) is standard in a statistics class when discussing linear regression."
But almost completely useless when doing numeric computations on numeric data.
In much the same way beginners use the determinent to test if a matrix is singular or not, because that is what they were taught in "statistics class", but in the real world of numeric computing: almost completely useless.
"But A'A is not necessarily invertible (although I have never encoutered a linear regression problem where it's not)."
Whether A'A is invertible is not really the problem here, this still avoids the numeric issue.
"So maybe Matlab has a way to deal with that possibility, by not using inv() when it does left matrix divide."
The MLDIVIDE() documentation explains what algorithms it uses. Read it.

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More Answers (2)

Steven Lord
Steven Lord on 20 Jun 2022
A=[4,12;6,8];
b=[6,12;14,8];
From the documentation for mrdivide, /, if x = A/b then x*b should be close to A.
x1 = A/b
x1 = 2×2
1.1333 -0.2000 0.5333 0.2000
check = x1*b-A
check = 2×2
0 0 0 0
Similarly from the documentation for mldivide, \, if x = A\b then A*x should be close to b.
x2 = A\b
x2 = 2×2
3.0000 0 -0.5000 1.0000
check = A*x2 - b
check = 2×2
1.0e-15 * 0.8882 0 0 0

John D'Errico
John D'Errico on 20 Jun 2022
BOTH of them are linear algebraic solutions. Where matrices are involved, they solve subtly different problems.
A\b solves the linear algebra problem A*X=b.
For these matrices...
A=[4,12;6,8];
b=[6,12;14,8];
X1 = A\b
X1 = 2×2
3.0000 0 -0.5000 1.0000
Did it work?
A*X1
ans = 2×2
6.0000 12.0000 14.0000 8.0000
Did it recover the matrix b? Yes.
What does forward slash do? Again, when matrices are involved, it solves a different problem. A/b is equivalent to solving the linear algebra problem X2*b=A.
X2 = A/b
X2 = 2×2
1.1333 -0.2000 0.5333 0.2000
Did it work?
X2*b
ans = 2×2
4 12 6 8
Essentially, the two are similar in philosophy. The difference is where the unknown matrix would be in the problem you are implicitly solving.

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