What is the transpose of a matrix $i[A,B]$?

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  • #1
Slimy0233
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Homework Statement
Q.6. If A and B are Hermitian matrices, then which of the following matrices will
be neither Hermitian nor skew hermitian in nature?
Relevant Equations
For a Hermitian Matrix A,
1. $A^{\dagger} = A$
2. $iA$ is a skew-Hermitian Matrix
1693057590678.png

I don't know how to simplify option b)

I mean, I don't know how to take the transpose of a matrix which exists as a product of $\iota$, how can I solve this further?

I need help only with option c. I have attempted this previously and have gotten what seems to be the right answer, but I don't know how I did that now
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  • #2
In general your solutions have a step missing. It's not a major problem, but you leave a bit of work for the reader.

You also need to assume that ##A, B \ne 0##.
 
  • #3
PeroK said:
In general your solutions have a step missing. It's not a major problem, but you leave a bit of work for the reader.

You also need to assume that ##A, B \ne 0##.
except for opt B and opt D you don't need extra steps as you can use property
The sum of two Hermitian Matrices is also a Hermitian matrix. But, I still get how to solve c though. I want to take the transpose of i[A,B], how can I do this?
 
  • #4
Slimy0233 said:
I want to take the transpose of i[A,B], how can I do this?
I thought you ready did in your OP.
 
  • #5
PeroK said:
I thought you ready did in your OP.
I didn't unfortunately, I did try and do it now which is attached, but I am pretty sure there's an easier way as I am not supposed to do this in the exams.
DocScanner 26-Aug-2023 22-28.jpg
 
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  • #6
That's okay. Perhaps it could be simpler, but all the steps are correct.
 
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  • #7
For example, you could first show that ##[A,B]## is skew-Hermitian, hence ##i[A, B]## is Hermitian.
 
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  • #8
PeroK said:
For example, you could first show that ##[A,B]## is skew-Hermitian, hence ##i[A, B]## is Hermitian.
Thank you! of c! I forgot the fact that [A, B] can also be treated like a matrix (even though it doesn't look like a conventional matrix)
This is what I was looking for
 
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  • #9
PeroK said:
For example, you could first show that ##[A,B]## is skew-Hermitian, hence ##i[A, B]## is Hermitian.
if A is skew-Hermitian then ##(iA)^{\dagger} = iA## making ##iA## skew-Hermitian$$(iA)^\dagger = i^*A^\dagger = -iA^\dagger = -i (-A) = iA$$
edit: I corrected something stupid I said
 
  • #10
You meant if ##A## is skew-Hermitean than ##\mathrm{i} A## is Hermitean, and that's in fact what you correctly have proven.
 
  • #11
If i may ask what the i stands for in
i[a,b]? I guess [a,b] is the conmutator?
 
  • #12
WWGD said:
If i may ask what the i stands for in
i[a,b]? I guess [a,b] is the conmutator?
I'd be willing to bet it's the imaginary unit...
 
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  • #13
Mark44 said:
I'd be willing to bet it's the imaginary unit...
So you're multiplying the conmutator by the imaginary unit?
 
  • #14
WWGD said:
So you're multiplying the conmutator by the imaginary unit?
Commutators are a topic I've never studied, but from the work shown, i is multiplying the result of the commutator action.
 
  • #15
Mark44 said:
Commutators are a topic I've never studied, but from the work shown, i is multiplying the result of the commutator action.
I _am_ multiplying. ;).
 
  • #16
WWGD said:
I _am_ multiplying. ;).
You may be, but so is ##i##. :oldbiggrin:
 
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  • #17
Mark44 said:
Commutators are a topic I've never studied, but from the work shown, i is multiplying the result of the commutator action.
It's standard stuff, as far as QM is concerned.
 
  • #18
PeroK said:
It's standard stuff, as far as QM is concerned.
Which is above my level at this point.
 

FAQ: What is the transpose of a matrix $i[A,B]$?

What is the transpose of a matrix?

The transpose of a matrix is an operation that flips a matrix over its diagonal, switching the row and column indices of each element. If you have a matrix \( A \), its transpose is denoted as \( A^T \) or \( A' \), and the element at the \( i \)-th row and \( j \)-th column of \( A \) becomes the element at the \( j \)-th row and \( i \)-th column of \( A^T \).

What does \( i[A,B] \) represent in matrix notation?

In matrix notation, \( i[A,B] \) typically represents the product of the imaginary unit \( i \) and the commutator of matrices \( A \) and \( B \). The commutator \( [A,B] \) is defined as \( AB - BA \), where \( A \) and \( B \) are matrices. Therefore, \( i[A,B] = i(AB - BA) \).

How do you find the transpose of \( i[A,B] \)?

To find the transpose of \( i[A,B] \), where \( [A,B] = AB - BA \), you use the properties of the transpose operation. The transpose of a product of matrices \( AB \) is \( (AB)^T = B^T A^T \), and the transpose of a difference is the difference of the transposes. Therefore, \( (i[A,B])^T = (i(AB - BA))^T = i((AB - BA)^T) = i(B^T A^T - A^T B^T) = i[B^T, A^T] \).

Is the transpose of \( i[A,B] \) equal to \( i[A,B] \)?

No, the transpose of \( i[A,B] \) is generally not equal to \( i[A,B] \). Instead, the transpose of \( i[A,B] \) is \( i[B^T, A^T] \). This follows from the properties of the transpose operation and the definition of the commutator.

Why is the transpose of \( i[A,B] \) given by \( i[B^T, A^T] \)?

The transpose of \( i[A,B] \) is given by \( i[B^T, A^T] \) because of the linearity and distributive properties of the transpose operation. When you take the transpose of \( i(AB - BA) \), you apply the transpose to each term separately and reverse the order of multiplication, resulting in \( i(B^T A^T - A^T B

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