Prove: (AB)*=A*B* Prove Matrix Conjugate Equality

[RJLiberator]

Homework Statement

Show that (AB)*=A*B*

Homework Equations

* = complex conjugate.
A,B = Matrices, A is an nxm matrix and B is a mxl matrix.

The Attempt at a Solution

Okay, last problem on this large, lovely homework assignment.

I feel like there's two general ways this homework has gone.

Either 1) Use summation notation to look at elements of the matrices and find that the components on each side are equal.

OR
2) Use complex properties to prove the statements.

I want to use route 2) as it is easier and more beautiful.

But I've been staring at this, seemingly simple, statement for a while now and can't check my first move.
It seems like an obvious statement.

If I let C = AB, and say c is within the complex numbers.
Then C* = the conjugate of C.

But this isn't what I want to prove, methinks.

 

[fzero]

Perhaps it would help to show first that you can write a complex matrix as a linear combination of real matrices, in analogy with what we can do with numbers.

 

[Dick]

Homework Statement

Show that (AB)*=A*B*

Homework Equations

* = complex conjugate.
A,B = Matrices, A is an nxm matrix and B is a mxl matrix.

The Attempt at a Solution

Okay, last problem on this large, lovely homework assignment.

I feel like there's two general ways this homework has gone.

Either 1) Use summation notation to look at elements of the matrices and find that the components on each side are equal.

OR
2) Use complex properties to prove the statements.

I want to use route 2) as it is easier and more beautiful.

But I've been staring at this, seemingly simple, statement for a while now and can't check my first move.
It seems like an obvious statement.

If I let C = AB, and say c is within the complex numbers.
Then C* = the conjugate of C.

But this isn't what I want to prove, methinks.

Now you should write the matrix product as a summation. And then use that $(ab)^*=a^*b^*$ for complex numbers.

 

[RJLiberator]

Oh man, that's an easy one!

You write the summation out, and use the fact that you are now dealing with real numbers and can use that property and boom!

How do you write summation notation in this scenario?

Here's what I did:

1. \left( \sum_{k=0}^m(a_{ij}b_{ij})^*\right)
2. \left( \sum_{k=0}^ma_{ij}^*b_{ij}^*\right)
3. =(a*b*)_ij
4. = A*B*

Step 1 is component notation
Step 2 is property of complex numbers
Step 3+4 bring it home.

 

[Dick]

Oh man, that's an easy one!

You write the summation out, and use the fact that you are now dealing with real numbers and can use that property and boom!

How do you write summation notation in this scenario?

Here's what I did:

1. \left( \sum_{k=0}^m(a_{ij}b_{ij})^*\right)
2. \left( \sum_{k=0}^ma_{ij}^*b_{ij}^*\right)
3. =(a*b*)_ij
4. = A*B*

Step 1 is component notation
Step 2 is property of complex numbers
Step 3+4 bring it home.

You are summing on an index, k, that isn't even in the expression you are summing. Look up the right way to express matrix multiplication in index form, ok?

 

[RJLiberator]

Er, for some reason I did it right in my homework, but wrong on here. Probably too much focus on the latex.

K=1 to m.
a_ik
b_kj

 

[Dick]

Er, for some reason I did it right in my homework, but wrong on here. Probably too much focus on the latex.

K=1 to m.
a_ik
b_kj

That's better.

 

[Fredrik]

Right, matrix multiplication is defined by $(AB)_{ij}=\sum_k A_{ik}B_{kj}$ and the adjoint is defined by $(C^*)_{ij}=(C_{ji})^*$. (The asterisk on the right denotes complex conjugation. You may prefer the notation $\overline C_{ji}$).

I'm not sure what most linear algebra books call the matrix that I called the adjoint, but I hope they don't call it the "complex conjugate", because that would be very misleading. C* denotes the transpose of the matrix that you get when you take the complex conjugate of each element of a matrix C.