Showing a 6x6 matrix has at least one positive eigenvalue

In summary, to show that a 6x6 matrix A with a negative determinant has at least one positive eigenvalue, one can sketch the graph of the characteristic polynomial of A, which is proportional to λ^6 and shifted by the determinant of A. By observing the value of the polynomial at λ=0 and for large positive λ, it can be shown that there will be at least one positive eigenvalue.
  • #1
macaholic
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Homework Statement



Show that if a 6x6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph for the characteristic polynomial of A.

Homework Equations


Characteristic polynomial: [itex](-\lambda)^n + (\text{tr}A)(-\lambda)^{n-1} + ... \text{det} A[/itex]


The Attempt at a Solution


I'm not really sure what to do at all. I know that the characteristic polynomial for a 6x6 matrix is going to be proportional to [itex]\lambda ^6[/itex], and shifted by det(A), and that the roots of the polynomial are going to be the eigenvalues...But I don't see how this shows there will be at least one positive eigenvalue. Can anyone point me in the right direction here?
 
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  • #2
What is the value of the polynomial at λ=0? For large positive λ?
 

Related to Showing a 6x6 matrix has at least one positive eigenvalue

1. How do you show that a 6x6 matrix has at least one positive eigenvalue?

To show that a 6x6 matrix has at least one positive eigenvalue, we can use the Perron-Frobenius theorem. This theorem states that for any non-negative matrix, the largest eigenvalue must be positive. Therefore, we can find the largest eigenvalue of the 6x6 matrix and prove that it is positive.

2. Can you explain the concept of eigenvalues and how they relate to matrices?

Eigenvalues are a characteristic of a matrix that represent the scale factor by which a vector is stretched or compressed when multiplied by that matrix. They are important because they can tell us about the behavior and properties of a matrix, such as whether it is invertible or what types of transformations it performs on vectors.

3. Are there any other methods for proving that a matrix has a positive eigenvalue?

Yes, there are other methods for proving that a matrix has a positive eigenvalue. Some other common techniques include using the Gershgorin circle theorem, determining the sign of the trace of the matrix, or using the intermediate value theorem.

4. Can a 6x6 matrix have more than one positive eigenvalue?

Yes, a 6x6 matrix can have more than one positive eigenvalue. The Perron-Frobenius theorem only guarantees that the largest eigenvalue will be positive, but there can be other positive eigenvalues as well. The exact number of positive eigenvalues will depend on the specific properties of the matrix.

5. Is it possible for a 6x6 matrix to have no positive eigenvalues?

Yes, it is possible for a 6x6 matrix to have no positive eigenvalues. This would mean that all of the eigenvalues are either zero or negative. In this case, the matrix may have other interesting properties, such as being nilpotent or having a determinant of zero.

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