What are the properties of complex and conjugate matrices in linear algebra?

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Hey I'd appreciate it if somebody could "yay or nay" my understanding of the following concepts, I think I'm mixing something up.

$$1. Complex Matrix$$
$$A = \begin{bmatrix} 1 & 1 + i & 2 + i \\ 2 + 3i & 4 - i & 3 + 2i \end{bmatrix}$$

$$2. Complex Conjugate Matrix$$
$$\overline{A} = \begin{bmatrix} 1 & 1 - i & 2 - i \\ 2 - 3i & 4 + i & 3 - 2i \end{bmatrix}$$

$$3. Transpose Matrix$$
$$A^T = \begin{bmatrix} 1 & 2 - 3i \\ 1 - i & 4 + i\\ 2 - i & 3 - 2i \end{bmatrix}$$

$$4. Conjugate Transpose Matrix$$
$$\overline{A}^T = \begin{bmatrix} 1 & 2 + 3i \\ 1 + i & 4 - i\\ 2 + i & 3 + 2i \end{bmatrix}$$

I think these last two may be confused or maybe even the 4th one is just my invention.
I'd appreciate a word of comment should any corrections be required.

5: What does $$A^*$$ usually signify, is it the complex conjugate or the complex transpose (assuming it exists).

 

[jbunniii]

Your (3) and (4) should be swapped. $A^T$ usually means just the transpose, not the conjugate transpose, even for a matrix with complex entries. (Some authors do mean conjugate transpose when they write $A^T$, though, so watch out for the context.)

As far as I know, $A^*$ pretty much universally means "conjugate transpose", not just "conjugate," although it would not completely surprise me if there is some eccentric author who uses it that way.

Another notation for conjugate transpose is $A^H$, where "H" stands for "Hermitian transpose," which means exactly the same thing as "conjugate transpose."

Matlab uses $A'$ for conjugate transpose, but I've never seen that notation used anywhere else.

 

[sponsoredwalk]

Whenever a matrix contains complex entries, the operation of complex conjugation
almost always accompanies the transpose operation.
(Recall that the complex conjugate of z = a + ib is
defined to be $$\overline{z}$$ = a - ib.)

---------------------------------From My Book
Thanks for the reply, yes it may be the context. From this quote & from the context would you now say I'm right in what I've previously written or am I still confused?

What should I switch in 3 & 4 (assuming it's still wrong), does the title of 3 represent the 4th matrix and vice versa or is everything okay considering the context this author has chosen?

I was mainly concerned that I had invented an extra matrix or something.

 

[jbunniii]

---------------------------------From My Book
Thanks for the reply, yes it may be the context. From this quote & from the context would you now say I'm right in what I've previously written or am I still confused?

I agree that the quote is a little vague. I think what the author means is that the conjugate transpose is the "natural" transpose of a complex matrix, i.e., most theorems/results that involve using the transpose of a real matrix require that you use a conjugate transpose if the matrix is complex.

[Something similar is true for plain old numbers. For example, consider the magnitude squared of a real number x. You can calculate it simply as $|x|^2 = x^2 = xx$. But for a complex number z, you need a complex conjugate on one of the factors: $|z|^2 = \overline{z}z$.]

The notation for a conjugate transpose is usually $A^*$ or $A^H$, and it's not a good idea to write $A^T$ when you mean conjugate transpose. I don't think that is what the author is suggesting (I hope it isn't), but like I said, it's a little vague.

P.S. Regarding what you should swap... if you swap just the matrices in (3) and (4), but leave the titles and the notation alone, then it will be correct, with the proviso that while $$\overline{A}^T$$ is perfectly valid notation for the conjugate transpose, it's much more usual to write $A^*$ or $A^H$.

 

[sponsoredwalk]

Your message helped me understand the book better. Thanks I've cracked it now,

$$A = [a_i_j]$$
$$\overline{A} = [\overline{a}_i_j]$$
$$A^T = [a_j_i]$$
$$\overline{A}^T = A^* = [\overline{a}_j_i]$$

I forget the exact Hermitian Matrix since I learned it so long ago & am relearning Lin Alg but it's up next so I'll be back assuming any problems :tongue2:

Cheers, have a good evening