Conjugate transpose/real and imaginary parts

In summary, the conversation discusses the definition of the real and imaginary parts of a complex matrix A and whether the given formulas for A1 and A2 are reasonable. There is no specific convention for defining the real and imaginary parts of a complex entity, but the given formulas involve complex entries and may not be considered reasonable. The formulas are trying to express the real and imaginary parts of each element of the matrix, but there is some disagreement among the posters about what should be considered the real and imaginary parts of a matrix.
  • #1
zcd
200
0
In my linear algebra text it says it's possible to define (for nxn matrix A)
[tex]A_1^* =\frac{A+A^*}{2}[/tex]
[tex]A_2^* =\frac{A-A^*}{2i}[/tex]
so A=A1+iA2

It then asked if this was a reasonable way to define the real and imaginary parts of A. Is there a specific convention to define the real and imaginary parts of something complex? It seems as if this way still contains complex entries in the Ai, so my guess is that it's not reasonable, but I want to make sure.
 
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  • #2
Resonable iff A is symmetric.
 
  • #3
But what would qualify as "reasonable"? That's my main question.
 
  • #4
Actually there is no definition about the real/imaginary parts of a matrix,
resonable means proper.
 
  • #5
Think about the real and imaginary parts of each element of the matrix.

[tex]a_{jk} = x_{jk} + i y_{jk}[/tex]

[tex]a^*_{jk} = x_{jk} - i y_{jk}[/tex]

[tex](a_{jk} + a^*_{jk} ) / 2 = x_{jk}[/tex]

[tex](a_{jk} - a^*_{jk} ) / 2i = y_{jk}[/tex]

That's all the formulas are trying to say.

I don't know what posts #2 and #4 are talking about.
 
  • #6
AlephZero said:
Think about the real and imaginary parts of each element of the matrix.

[tex]a_{jk} = x_{jk} + i y_{jk}[/tex]

[tex]a^*_{jk} = x_{jk} - i y_{jk}[/tex]

[tex](a_{jk} + a^*_{jk} ) / 2 = x_{jk}[/tex]

[tex](a_{jk} - a^*_{jk} ) / 2i = y_{jk}[/tex]

That's all the formulas are trying to say.

I don't know what posts #2 and #4 are talking about.
I think it's more likely the original poster was using * for conjugate transpose, rather than for the complex conjugate.

Posts #2 and #4 suggest that Some Pig has decided what you wrote is the only reasonable meaning for "real part of a matrix", and your formula only agrees with the opening post's formula in the case that A is symmetric.



IMO, the role that A1 and A2 plays in the matrix algebra is much closer in spirit to the role that real and imaginary parts play for complex numbers than the matrices you suggest, and IMO the main obstacle to the reasonability of calling them the real and imaginary parts are the likelihood that people would think of the matrices you have defined, rather than the matrices of the opening post. That A1 and A2 are not matrices over the reals is also an obstacle, but IMO a rather small one.
 

Related to Conjugate transpose/real and imaginary parts

What is the difference between the conjugate transpose and the regular transpose?

The conjugate transpose of a matrix is obtained by taking the transpose of the matrix and then replacing each element with its complex conjugate. This means that the real and imaginary parts of each element are flipped. On the other hand, the regular transpose simply flips the rows and columns of a matrix without changing the values of the elements.

What are the applications of the conjugate transpose?

The conjugate transpose is commonly used in linear algebra and signal processing. It is particularly useful in solving systems of linear equations, calculating eigenvalues and eigenvectors, and in defining unitary and Hermitian matrices.

What are the real and imaginary parts of a complex number?

A complex number can be written in the form a + bi, where a is the real part and bi is the imaginary part. The real part represents the horizontal axis on the complex plane, while the imaginary part represents the vertical axis.

How are the real and imaginary parts related to the conjugate of a complex number?

The conjugate of a complex number is obtained by changing the sign of the imaginary part. Geometrically, this means reflecting the complex number across the real axis on the complex plane. The conjugate of a complex number a + bi is written as a - bi.

What is the relationship between the conjugate transpose and the Hermitian transpose?

The conjugate transpose and the Hermitian transpose are two different names for the same operation. Both terms refer to taking the transpose of a matrix and then replacing each element with its complex conjugate. In other words, the Hermitian transpose is the same as the conjugate transpose.

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