Can I prove matrix properties using simple steps?

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Homework Help Overview

The discussion revolves around proving properties of matrices, specifically focusing on the commutator of two matrices, X and Y. Participants are exploring whether they can use general notation or if they need to define specific matrices to demonstrate the properties effectively.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the validity of using general matrix notation versus defining specific matrices. There is an exploration of the expressions involving the commutator [X, Y] and its properties, with some questioning the clarity of notation used.

Discussion Status

Some participants have provided guidance on the notation and the steps involved in manipulating the expressions. There appears to be a productive exchange regarding the validity of the steps taken, though no consensus has been reached on the necessity of defining specific matrices.

Contextual Notes

There is confusion regarding the notation and the representation of matrices, with participants noting that uppercase letters denote matrices while lowercase letters represent matrix entries. The discussion also reflects on the complexity of proving properties of matrices and the need for clarity in notation.

joedozzi
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Homework Statement


Homework Equations



Question 1.jpg



The Attempt at a Solution



-(y, x) = -(YX-XY)
= XY-YX

Can I do this or would I have to define a matrix X= ( a b c d ) Y= ( e f g h)

And prove it that way? I am just really confused
 
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joedozzi said:

Homework Statement


Homework Equations



View attachment 50857


The Attempt at a Solution



-(y, x) = -(YX-XY)
= XY-YX

Can I do this or would I have to define a matrix X= ( a b c d ) Y= ( e f g h)

And prove it that way? I am just really confused

What you have above would work, but your notation is awful! Try to use the notation as given in the problem. Also notice that uppercase letters represent matrices, and lowercase letters represent the entries in a matrix.

Assuming that matrices X and Y are in M, then [X, Y] = ?
Keep working with the expressions you get until you end up with -[Y, X].
 
[X, Y]= YX- XY
Thus [Y, X]= XY- YX
then -[Y, X]= -(XY- YX)
then -[Y, X]= -XY +YX

Like that? So i don't have to use matricies with variables, just use X and Y to represent a matrix?
 
joedozzi said:
[X, Y]= YX- XY
Thus [Y, X]= XY- YX
then -[Y, X]= -(XY- YX)
then -[Y, X]= -XY +YX

Like that? So i don't have to use matricies with variables, just use X and Y to represent a matrix?

This is easier to follow.

[X, Y] = XY - YX = -(YX - XY) = -[Y, X]

Can you say why each pair of successive equal expressions is valid?
 
Properties of Matricies?
 
And Thanks your honestly a huge help!
 
joedozzi said:
Properties of Matricies?
That's pretty vague. Also, there are a number of steps. One reason doesn't fit them all.
 

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