Eigenvalues of 2 matrices are equal

In summary, the conversation is about trying to prove that two eigenvalues of matrices A and B are equal. The speaker attempted to solve the problem using determinants but the result was not complete. They are asking for help in getting the complete result.
  • #1
gopi9
14
0
Hi all,

I have two matrices
A=0 0 1 0
0 0 0 1
a b a b
c d c d
and B=0 0 0 0
0 0 0 0
0 0 a b
0 0 c d
I need to prove that two eigenvalues of A and two eigenvalues of B are equal. I tried to take the determinant of A-λI and B-λI and solve them but the result is not complete, the result that I got is
if e,f,g,h are eigenvalues of A and i,j,k,l are eigenvalues of B then
e+f+g+h=a+d;
i+j=a+d, k=l=0;
e*f*g*h=i*j;
efg+fgh+efh+egh=-2ij

Can anyone get me the complete result that is two eigenvalues of A and B are equal?

Thanks
 
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Related to Eigenvalues of 2 matrices are equal

1. What are eigenvalues?

Eigenvalues are numbers that are associated with a square matrix. They represent the scaling factor by which a vector is stretched or compressed when multiplied by the matrix.

2. Can two matrices have the same eigenvalues?

Yes, it is possible for two matrices to have the same eigenvalues. This means that the two matrices have the same impact on the vectors that are multiplied by them.

3. How can we determine if two matrices have equal eigenvalues?

To determine if two matrices have equal eigenvalues, we can calculate the characteristic polynomial for each matrix and compare them. If the polynomials are the same, then the matrices have equal eigenvalues.

4. What does it mean if two matrices have equal eigenvalues?

If two matrices have equal eigenvalues, it means that the two matrices have the same impact on the vectors that are multiplied by them. This can provide insights into the relationship between the two matrices.

5. Is it possible for two matrices to have equal eigenvalues but be different matrices?

Yes, it is possible for two matrices to have equal eigenvalues but be different matrices. This means that while the two matrices have the same impact on vectors, they may have different structures and may affect other calculations differently.

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