Trace(matrix) = 0 and the dimension of subspace

In summary, the conversation discusses the claim that matrices of trace zero form a subspace of M_n (F) of dimension n^2 - 1. The trace is viewed as a linear transformation and the dimension of the subspace is found using the fact that dim M_n (F) = dim Tr [M_n (F)] + dim null [M_n (F)]. It is clarified that the dimension of the subspace is not 0, but rather 1, and the subspace is generated by the matrices A, B, and their commutator.
  • #1
Ioiô
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0

Homework Statement



Claim: Matrices of trace zero for a subspace of M_n (F) of dimension n^2 -1 where M_n (F) is the set of all nxn matrices over some field F.

Homework Equations



Tr(M_n) = sum of diagonal elements

The Attempt at a Solution



I view the trace Tr as a linear transformation Tr: M_n (F) -> F. I find the dimension of the subspace spanned by M_n (F) by using the fact:

dim M_n (F) = dim Tr [M_n (F)] + dim null [M_n (F)]

Since dim M_n (F) = n^2, we have

n^2 = dim Tr [M_n (F)] + dim null [M_n (F)]

I'm confused on what's in the right part of the above equation:

Isn't dim Tr [M_n (F)] = 0 since Tr [M_n (F)] = 0?

Shouldn't dim null [M_n (F)] =n^2 - dim Tr [M_n (F)] = n^2 - 1?

Also, I know that the subspace is generated by the following matrices: A, B, and [A,B] (the commutator of A and B).

Thanks!
 
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  • #2
You said

> Isn't dim Tr [M_n (F)] = 0 since Tr [M_n (F)] = 0?

No, dim Tr [M_n (F)] = 1 since Tr [M_n (F)] = F
You have confused M_n (F) (all nxn matrices over F) with its subset of trace zero matrices. And a (non degenerate) field over itself obviously has dimension 1 because it's spanned by its element 1 (multiplicative identity). So you get indeed

dim null [M_n (F)] =n^2 - dim Tr [M_n (F)] = n^2 - 1
 

1. What does it mean for a matrix to have a trace of 0?

The trace of a matrix is the sum of its diagonal entries. If a matrix has a trace of 0, it means that the sum of its diagonal entries is equal to 0.

2. How does the trace of a matrix relate to the dimension of a subspace?

The trace of a matrix and the dimension of a subspace are related through the rank-nullity theorem. The rank of a matrix is equal to the dimension of its column space, and the nullity is equal to the dimension of its null space. If the trace of a matrix is 0, it means that the sum of the dimensions of its column space and null space is equal to the total number of columns, which is the dimension of the entire space.

3. Can a matrix have a trace of 0 and still have a non-zero determinant?

Yes, it is possible for a matrix to have a trace of 0 and a non-zero determinant. For example, a diagonal matrix with all entries equal to 0 except for one entry on the diagonal with a non-zero value would have a trace of 0 but a non-zero determinant.

4. How can the trace of a matrix help in determining its eigenvalues?

The trace of a matrix is equal to the sum of its eigenvalues. Therefore, if a matrix has a trace of 0, it means that the sum of its eigenvalues is also 0. This can aid in finding the eigenvalues of a matrix through methods such as the characteristic polynomial or the diagonalization method.

5. Is the trace of a matrix always equal to the dimension of the subspace it represents?

No, the trace of a matrix is not always equal to the dimension of the subspace it represents. The trace only provides information about the dimension of the column space and null space of a matrix, not the entire subspace. Additionally, the dimension of a subspace can be greater than the trace of a matrix.

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