Matrices: Question about Commutativity

  • Thread starter Thread starter flash
  • Start date Start date
  • Tags Tags
    Matrices
AI Thread Summary
The discussion centers on the commutativity of matrix operations involving scalars and quaternions. It confirms that scalars can be moved freely in matrix multiplication, allowing the expression X * kY to be simplified to k(X * Y). The conversation highlights the difference between commutative and non-commutative rings, specifically mentioning quaternions. The user is working with complex matrices and aims to demonstrate that Q * Q-1 equals the identity matrix, successfully applying the discussed principles. Ultimately, the manipulation of scalars in this context is affirmed as valid and effective.
flash
Messages
66
Reaction score
0
I have a question about commutativity.
I have two matrices X and Y and a constant k. I want to calculate X * kY. Can I bring k out the front to give k(X*Y)?
 
Physics news on Phys.org
Thanks for the link. It says "When the underlying ring is commutative, for example, the real or complex number field, the two multiplications are the same. However, if the ring is not commutative, such as the quaternions, they may be different."

Lol, I am actually working with quaternions. The matrices contain complex numbers. I am trying to show that
Q * Q-1 = Identity
but Q-1 is kX because its the inverse of a 2x2 matrix. So I thought it would be easier to work out QX then multiply the answer by k to (hopefully) give the Identity matrix (if that makes any sense at all).
 
Well then, I bow to people more knowledgeable than I about manipulating quarternions.
 
Same statement is true: a scalar (number), k, can be moved around pretty much as you wish. It is only multiplication of the matrices or quaternions that is non-commutative.
 
Thanks for the help, I moved k out the front and it worked :smile:
 

Thread 'Family of lines that are at a distance of 5 from the origin'

I picked up this problem from the Schaum's series book titled "College Mathematics" by Ayres/Schmidt. It is a solved problem in the book. But what surprised me was that the solution to this problem was given in one line without any explanation. I could, therefore, not understand how the given one-line solution was reached. The one-line solution in the book says: The equation is ##x \cos{\omega} +y \sin{\omega} - 5 = 0##, ##\omega## being the parameter. From my side, the only thing I could...
View full post »

Thread 'Making the denominator of a certain fraction real'

Essentially I just have this problem that I'm stuck on, on a sheet about complex numbers: Show that, for ##|r|<1,## $$1+r\cos(x)+r^2\cos(2x)+r^3\cos(3x)...=\frac{1-r\cos(x)}{1-2r\cos(x)+r^2}$$ My first thought was to express it as a geometric series, where the real part of the sum of the series would be the series you see above: $$1+re^{ix}+r^2e^{2ix}+r^3e^{3ix}...$$ The sum of this series is just: $$\frac{(re^{ix})^n-1}{re^{ix} - 1}$$ I'm having some trouble trying to figure out what to...
View full post »

Similar threads

Show that the ratio ##x+y:x-y## is increased by subtracting ##y##
Replies
4
Views
1K
Identifying matrices as REF, RREF, or neither
Replies
1
Views
3K
Prove the identity matrix is unique
2
Replies
69
Views
8K
Transformations to both sides of a matrix equation
Replies
25
Views
2K
Calculating the dimension of intersection of two matrices
Replies
7
Views
3K
How Do Hadamard Matrices Relate to Identity Matrices?
Replies
2
Views
2K
I Variant of Baker-Campbell-Hausdorff Formula
Replies
3
Views
2K
Solving Sets of Matrices for Proving Equivalence Relation
Replies
8
Views
2K
Using matrices for functions -- transformations and translation
Replies
13
Views
2K
A Question about commutator involving fermions and Pauli matrices
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top