Conjugate transpose/real and imaginary parts

[zcd]

In my linear algebra text it says it's possible to define (for nxn matrix A)
A_1^* =\frac{A+A^*}{2}
A_2^* =\frac{A-A^*}{2i}
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 A_i, so my guess is that it's not reasonable, but I want to make sure.

 

[Some Pig]

Resonable iff A is symmetric.

 

[zcd]

But what would qualify as "reasonable"? That's my main question.

 

[Some Pig]

Actually there is no definition about the real/imaginary parts of a matrix,
resonable means proper.

 

[AlephZero]

Think about the real and imaginary parts of each element of the matrix.

a_{jk} = x_{jk} + i y_{jk}

a^*_{jk} = x_{jk} - i y_{jk}

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

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

That's all the formulas are trying to say.

I don't know what posts #2 and #4 are talking about.

 

[Hurkyl]

Think about the real and imaginary parts of each element of the matrix.

a_{jk} = x_{jk} + i y_{jk}

a^*_{jk} = x_{jk} - i y_{jk}

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

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

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.