Is Rank(A) Equal to Trace(AA*) and When Does AB-BA=I Hold?

[arthurhenry]

Rank(A)= Trace(AA*) ??

I have two questions and I hope it is acceptable...Seemingly unrelated, though I came to wonder about the first while thinking the second. Thanks

1)Is this statement true? or is there a statement that relates Rank(A) and Trace(??)

2) AB-BA=I (When does this identity hold if at all? Field can be closed or Z/Z2, or etc)

 

[micromass]

Hi arthurhenry!

The first one is not true. It is true that Rank(A)=Rank(AA*). But it isn't in general true that Rank(AA*)=Trace(AA*). For example, take

$$A=\left(\begin{array}{cc} 2 & 0\\ 0 & 2\\ \end{array}\right)$$

Then AA* has rank 2, but the trace is 8.

The second one is not true too. Take A the zero matrix and B an arbitrary matrix.

 

[arthurhenry]

Perhaps I was not clear, I will phrase it correctly:

2)Does there exist matrices A and B such that AB-BA=I holds?

In particular, what is the answer to the question in the case the field is Complex NUmbers and in the case the field is Z/Z2 ?

1) Is rank(A) equal to Trace(A*A) ?

not "is the rank(A*A) equal to Trace (A*A)?" As you have pointed out this one is definittely incorrect.

 

[micromass]

Perhaps I was not clear, I will phrase it correctly:

2)Does there exist matrices A and B such that AB-BA=I holds?

In particular, what is the answer to the question in the case the field is Complex NUmbers and in the case the field is Z/Z2 ?

Well, try it yourself. Take general 2x2-matrices and calculate AB-BA. Then solve the system to see whether they can equal I...

1) Is rank(A) equal to Trace(A*A) ?

not "is the rank(A*A) equal to Trace (A*A)?" As you have pointed out this one is definittely incorrect.

We have that rank(A)=rank(A*A), so the same example applies.

 

[blue_raver22]

1) The statement "Rank(A) = Trace(AA*)" is not necessarily true. The trace of a matrix is the sum of its eigenvalues, while the rank is the number of linearly independent columns (or rows) in the matrix. These two quantities are not directly related, so there is no statement that can relate them in general.

2) The identity AB-BA = I holds when A and B are inverse matrices. This means that AB = BA = I, and this can only happen if A and B are square matrices with non-zero determinants. Over the field of real or complex numbers, this can happen for any size of square matrices. However, over other fields such as Z/Z2, this may not hold for all sizes of square matrices.