Is A./B different from B.\A?
20 vues (au cours des 30 derniers jours)
Afficher commentaires plus anciens
Oliver Woodford
le 17 Juin 2015
Modifié(e) : James Tursa
le 29 Oct 2021
Given two matrices, A and B, will A./B ever give a different answer from B.\A, or are these two expressions equivalent?
It seems that even for complex numbers they return the same thing. E.g.
>> A = sqrt(randn(3));
>> B = sqrt(randn(3));
>> isequal(A./B, B.\A)
ans = 1
Réponse acceptée
James Tursa
le 17 Juin 2015
I can't think of any reason why one would ever get different results for numeric types. I suppose there might be speed differences if one form used multi-threading and the other form didn't, but in tests I just ran they both appeared to take about the same amount of time.
User defined classes could of course overload them differently.
6 commentaires
Bruno Luong
le 30 Sep 2021
Modifié(e) : Bruno Luong
le 30 Sep 2021
"The only way to get non-commutative objects such as quaternions is to create a class for them"
But they are available in some toolboxes, e.g., https://www.mathworks.com/help/robotics/ref/quaternion.html
I don't have any of those toolboxes to check how "A./B" and "B.\A" works, but I expect them NOT give the same results for general cases:
James Tursa
le 29 Oct 2021
Modifié(e) : James Tursa
le 29 Oct 2021
Yes, your expectations are correct. For the MATLAB toolbox quaternion class objects, the q./p and p.\q operations are implemented as expected by multiplying by the inverse, and since multiplication is non-commutative you get different results.
>> x = rand(1,4)-0.5; x = x/norm(x); q = quaternion(x);
>> x = rand(1,4)-0.5; x = x/norm(x); p = quaternion(x);
>> q
q =
quaternion
-0.62168 + 0.46748i + 0.58112j + 0.23933k
>> p
p =
quaternion
0.64169 + 0.60532i - 0.26832j + 0.38709k
>> q./p
ans =
quaternion
-0.17923 + 0.38713i + 0.24217j + 0.87141k
>> p.\q
ans =
quaternion
-0.17923 + 0.96545i + 0.17j - 0.082977k
>> q*conj(p)
ans =
quaternion
-0.17923 + 0.38713i + 0.24217j + 0.87141k
>> conj(p)*q
ans =
quaternion
-0.17923 + 0.96545i + 0.17j - 0.082977k
>> which quaternion
C:\Program Files\MATLAB\R2020a\toolbox\shared\rotations\rotationslib\@quaternion\quaternion.m % quaternion constructor
Note that the / and \ operators are not implemented for this class:
>> q/p
Error using /
Arguments must be numeric, char, or logical.
>> p\q
Error using \
Arguments must be numeric, char, or logical.
Plus de réponses (2)
Alberto
le 17 Juin 2015
Both are pointwise, but A./B divides every element in A by the same element in B. A.\B divides every element in B by the same element in A.
H. Sh. G.
le 28 Sep 2021
Hi every body.
I wonder what kind of calculations the division of a matrix (X) by a row vector (y), i.e. X/y, does, where both have the same number of columns.
The result is a column vector of the same number of rows that X has.
Recall that X./y divides all elements of each column in X by the element of y in the same column, resulting in a matrix with the same size of X.
4 commentaires
Bruno Luong
le 29 Sep 2021
Modifié(e) : Bruno Luong
le 30 Sep 2021
"Yes, I know that b/A gives pinv(A)*b."
It's incorrect. In some cases b/A is A*pinv(b) (and not the opposite as you wrote)
B=rand(2,4);
A=rand(3,4);
A/B
A*pinv(B)
but when rank(b) < size(b,1) such formula is not correct
B=rand(4,2);
A=rand(3,2);
A/B
A*pinv(B)
Now to your question.
In case B = b is a row vector, let consider the system
% x*B = A;
x is column vector (since is a row vector). This system works row by row of x and A independenty. So consider a row equation
% x(i) * b = A(i,:).
So what you ask is which scalar x(i) that when multiplying with a vector (b) must be equal to another vector A(i,:). Such solution does not exist unless A(i,:) is proportional to b. In general MATLAB returns the least square solution:
% x(i) = dot(A(i,:),b) / dot(b,b)
Illustration test :
b=rand(1,4);
A=rand(3,4);
A/b
A1=A(1,:); x1=dot(A1,b)/dot(b,b)
A2=A(2,:); x2=dot(A2,b)/dot(b,b)
A3=A(3,:); x3=dot(A3,b)/dot(b,b)
% Or all together
x=A*b'/(b*b')
Note that for a row vector b, pinv(b) is b'/(b*b').
And x*B will not match A, unless all the rows of A are proportional to b (which is a strong coincidence in general).
Voir également
Catégories
En savoir plus sur Quaternion Math dans Help Center et File Exchange
Community Treasure Hunt
Find the treasures in MATLAB Central and discover how the community can help you!
Start Hunting!