Can the trace be expressed in terms of the determinant?

In summary, the conversation discusses the relationship between determinants and eigenvalues and trace. The question posed is whether it is possible to express the determinant in terms of the trace, and vice versa. However, the conclusion is that while it is possible to express the determinant in terms of weighted sums of the trace, it is not possible to do the reverse due to the presence of products in the formulas.
  • #1
Jhenrique
685
4
Browsing in the wiki, I found those formulas:

26e7154aa1157002cd3db31b80531792.png

edcb9a111a6850c7b8efc7151807cc50.png

50c52e2460a73655bd0adc5a9ec6ac8e.png

http://en.wikipedia.org/wiki/Determinant#Relation_to_eigenvalues_and_trace

So, my doubt is: if is possible to express the determinant in terms of the trace, thus is possible to express the trace in terms of the determinant too?
 
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  • #2
I don't think so. We can write ##a_1\ldots a_n## in terms of weighted sums of ##(a_1+\ldots +a_n)^k## but I don't think the other way around. As in the Wikipedia article (and the formulas depend on the matrix sizes!), we can subtract what disturbs, but how should we get rid of products to achieve a sum?
 

1. What is the definition of trace in terms of determinant?

The trace of a square matrix is defined as the sum of the elements on its main diagonal. In terms of determinant, the trace of a matrix A is equal to the sum of its eigenvalues. This can be written as tr(A) = λ1 + λ2 + ... + λn, where λ1, λ2, ..., λn are the eigenvalues of A.

2. How is trace related to the determinant of a matrix?

The determinant of a square matrix is equal to the product of its eigenvalues. Therefore, the sum of the eigenvalues (trace) is related to the product of the eigenvalues (determinant) by the formula det(A) = λ1 * λ2 * ... * λn. This relationship holds for all square matrices.

3. Can trace be negative in terms of determinant?

No, trace cannot be negative in terms of determinant. Since the determinant is equal to the product of the eigenvalues, and the trace is equal to the sum of the eigenvalues, all eigenvalues must be positive for the determinant to be positive. Therefore, the trace must also be positive.

4. How is the trace of a matrix affected by elementary row operations?

Elementary row operations, such as multiplying a row by a constant or swapping two rows, do not affect the trace of a matrix. This is because the main diagonal elements, which determine the trace, remain the same after these operations. However, adding a multiple of one row to another can change the trace of a matrix.

5. Can the trace of a matrix be zero in terms of determinant?

Yes, the trace of a matrix can be zero in terms of determinant. This happens when all the eigenvalues of the matrix are zero. In this case, the determinant is also zero, as it is equal to the product of the eigenvalues. This is a special case, as the trace is usually a non-zero value.

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