Inverse Eigenvalues: A Puzzling Question?

In summary, the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^TA$, and this holds for any invertible matrix $A$.
  • #1
mathmari
Gold Member
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Hey! :eek:

Does it stand that the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^TA$ ?? (Wondering)
 
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  • #2
It is as followed, right??

The eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $(A^{-T}A^{-1})^{-1}=AA^{T}$ and the eigenvalues of $AA^{T}$ are the same as the eigenvalues of $A^{T}A$. So the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^{T}A$.

(Wondering)

But for which matrices $A$ does this stand?? (Wondering)
 
  • #3
mathmari said:
It is as followed, right??

The eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $(A^{-T}A^{-1})^{-1}=AA^{T}$ and the eigenvalues of $AA^{T}$ are the same as the eigenvalues of $A^{T}A$. So the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^{T}A$.

(Wondering)

But for which matrices $A$ does this stand?? (Wondering)
As long as $A$ is invertible, the proposition is valid as you have shown.
 

FAQ: Inverse Eigenvalues: A Puzzling Question?

What are inverse eigenvalues and why are they important?

Inverse eigenvalues refer to the reciprocal of the eigenvalues of a matrix. Eigenvalues are important in linear algebra and often used in solving systems of equations and analyzing the behavior of linear transformations.

How do you calculate inverse eigenvalues?

To calculate inverse eigenvalues, you first need to find the eigenvalues of the matrix. Then, simply take the reciprocal of each eigenvalue to get the inverse eigenvalues.

How do inverse eigenvalues relate to the original matrix?

Inverse eigenvalues are closely related to the original matrix. They can provide information about the behavior and properties of the matrix, such as its determinant, trace, and invertibility.

Are inverse eigenvalues always real numbers?

No, inverse eigenvalues can be complex numbers. This is because the eigenvalues of a matrix can be complex, and taking the reciprocal of a complex number will also result in a complex number.

Can you use inverse eigenvalues to solve systems of equations?

Yes, inverse eigenvalues can be used to solve systems of equations. They can help determine the stability and convergence of a system, which is useful in solving it. However, they are not the only factor to consider and should be used in conjunction with other methods.

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