conjugate transpose and matrix multiplication

[diogomgf]

Let A_m,n be any matrix

If $\displaystyle (A \cdot B)$ = $\displaystyle A^* \cdot B^*$ and $\displaystyle (A^*)^* = A$ ,
Then $\displaystyle (A^* \cdot A)^*$ = $\displaystyle (A^*)^* \cdot A^*$ = $\displaystyle A \cdot A^*$

But then also

$\displaystyle (A^* \cdot A)^* = \overline{(\overline{A^\top}\cdot A)^\top} = (A^* \cdot A)$

What am I doing wrong?

EDIT: Also, where can I find a comprehensive guide for the forum notations?

Thanks

 

[diogomgf]

Just wanted to edit this part here:

$\displaystyle (A \cdot B)^* = A^* \cdot B^*$

 

[Dr.Peterson]

Let A_m,n be any matrix

If $\displaystyle (A \cdot B)^*$ = $\displaystyle A^* \cdot B^*$ and $\displaystyle (A^*)^* = A$ ,
Then $\displaystyle (A^* \cdot A)^*$ = $\displaystyle (A^*)^* \cdot A^*$ = $\displaystyle A \cdot A^*$

But then also

$\displaystyle (A^* \cdot A)^* = \overline{(\overline{A^\top}\cdot A)^\top} = (A^* \cdot A)$

What am I doing wrong?

EDIT: Also, where can I find a comprehensive guide for the forum notations?

Thanks

First, what do you mean by "the forum notations"? We don't have any official notation, as far as I know; some notations vary around the world, and we have to interpret what people mean. In this case, what do YOU mean by [MATH]A^*[/MATH]? I note here that there are at least four different notations for the conjugate transpose, which I presume is what you mean. The standard thing to do is to state what you mean by your notation, whenever there is a good chance that others will not understand it. (And sometimes it takes some knowledge of context to recognize a notation you are entirely familiar with, which becomes mystifying out of context!)

I presume that what you are troubled by is the apparent implication that [MATH]A \cdot A^* = A^* \cdot A[/MATH], and you assume it isn't true that any matrix commutes with its own conjugate transpose?

But start at the beginning. Is it really true that [MATH](A \cdot B)^* = A^* \cdot B^*[/MATH]? (See the link above.)

 

[pka]

I have one additional question question to those of Prof. Peterson: What the heck does the transpose of a matrix, $\displaystyle A^T$, have anything to do with the discussion at hand?

 

[Romsek]

I have one additional question question to those of Prof. Peterson: What the heck does the transpose of a matrix, $\displaystyle A^T$, have anything to do with the discussion at hand?

$\displaystyle A^* = \overline{A}^T$

 

[pka]

$\displaystyle A^* = \overline{A}^T$

Will that is newer use than I am privy to, it is not in Perlis.

 

[Dr.Peterson]

Note my reference to the Wikipedia article on "conjugate transpose", which was identified as the subject of the question, for which one symbol is [MATH]A^*[/MATH], and there are also several alternative names. Yes, I had to look it up to verify all this; I have never taught linear algebra. (Maybe that makes me more open to varied notations and names than I would otherwise be.)

 

[Romsek]

Will that is newer use than I am privy to, it is not in Perlis.

it's not standard by any means but OP mentioned conjugate transpose in the subject line

 

[diogomgf]

First, what do you mean by "the forum notations"? We don't have any official notation, as far as I know; some notations vary around the world, and we have to interpret what people mean. In this case, what do YOU mean by [MATH]A^*[/MATH]? I note here that there are at least four different notations for the conjugate transpose, which I presume is what you mean. The standard thing to do is to state what you mean by your notation, whenever there is a good chance that others will not understand it. (And sometimes it takes some knowledge of context to recognize a notation you are entirely familiar with, which becomes mystifying out of context!)

The notations I'm refering to are the [/tex] notations, the LaTex ones. I can't find any comprehensive guide anywhere, just bits and tips around various different forums.

But start at the beginning. Is it really true that [MATH](A \cdot B)^* = A^* \cdot B^*[/MATH]? (See the link above.)

The book I am using to study linear algebra assumes it is, and therefore I also assume it is... It might be wrong.
The link you provided has it that: $\displaystyle (A \cdot B)^* = (B^* \cdot A^*).$

 

[Dr.Peterson]

Please quote what your book says, first about the definition of [MATH]A^*[/MATH], and then about [MATH](A \cdot B)^*[/MATH]. An image might be good.

Are you aware of the fact that [MATH](A \cdot B)^T = B^T \cdot A^T[/MATH]?

As for LaTeX, I would start with https://www.freemathhelp.com/forum/threads/latex-math-formatting-is-available.109843/.

 

[diogomgf]

Please quote what your book says, first about the definition of [MATH]A^*[/MATH], and then about [MATH](A \cdot B)^*[/MATH]. An image might be good.

An image won't help unless you can speak Portuguese but I'll do a rough summary:
First they start by saying that: $\displaystyle \overline{(A \cdot B)} = \overline{A} \cdot \overline{B}$ , among other conjugate matrix properties.
Then they tell us to show that the transcojugate matrix has the same properties.

Are you aware of the fact that [MATH](A \cdot B)^T = B^T \cdot A^T[/MATH]?

Yes, they have that definition.

As for LaTeX, I would start with https://www.freemathhelp.com/forum/threads/latex-math-formatting-is-available.109843/.

Thanks alot, didn't find this link on my research about LaTeX.

 

[Dr.Peterson]

An image won't help unless you can speak Portuguese but I'll do a rough translation:
First they start by saying that: $\displaystyle \overline{(A \cdot B)} = \overline{A} \cdot \overline{B}$ , among other properties.
Then they tell us to show that the transcojugate has the same properties.

Are you saying that the properties of the "transconjugate" are never actually stated explicitly? What does "the same properties" mean?

Note that the property we are discussing is parallel to the property of the transpose that you said they do state (though it is a property, not a definition). So that may count as "the same property", and would mean that they at least imply the property.

It would not be a bad idea to show us an image; you never know what languages someone may speak, or if they might be willing to look things up; seeing what equations are shown might be all we need.

 

[diogomgf]

1st image: Book cover.
2nd image: Preposition 1.37 - Properties of conjugate matrices.
3rd image: Definition of a transcojugate matrice.
4th image: Exercise 1.38 - Show that the properties of transcojugate matrices are the same as in proposition 1.37 (2nd image).

 

[Dr.Peterson]

Thanks.

I see that the exercise says (according to Google),

Exercise 1.38: Demonstrate, for the transconjugate, the properties corresponding to Proposition 1.37.​

Unless there is a solution in the back of the book, it appears that they have not clarified what "corresponding" means. It can't mean that you can just directly replace the conjugate with the transconjugate everywhere, as we know the transpose behaves differently. What they must mean is that you must make appropriate small changes in some of the properties (namely, #4) to produce "corresponding" (not "identical") properties -- in particular, we must prove only properties that are (provably) true, not just what we think they should be. You can't assume a property (apparently) implied by an exercise without having done that exercise to confirm what it is actually saying is true.

And the work you asked about is essentially a proof that the property you assumed is false.

 

[diogomgf]

You can't assume a property (apparently) implied by an exercise without having done that exercise to confirm what it is actually saying is true.

Well, the whole OP is about that last part. I actually assumed they were the same properties, we just needed to replace the conjugate with the transconjugate. I tried to prove this without success. Then I researched and found the second solution. I was confused and came here to double check.

Guess I was wrong in assuming the properties had to be exactly the same, but this books never have solutions for proof exercises.

Thanks once again for the clarification.