Eigenvalues of the power of a matrix

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  • #1
IniquiTrance
190
0

Homework Statement



If [itex]\lambda_i[/itex] are the eigenvalues of a matrix [itex]A^2[/itex], and [itex]A[/itex] is symmetric, then what can you say about the eigenvalues of A?

Homework Equations


The Attempt at a Solution



I know how to prove that if [itex]\sqrt(\lambda_i)[/itex] is an eigenvalue of A, then [itex]\lambda_i [/itex] is an eigenvalue of [itex]A^2[/itex].

I don't know how to prove the converse though.

Thanks!
 
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  • #2
Let [itex]\lambda[/itex] be an eigenvalue of [itex]A^2[/itex]. What can you say about [itex]A^2-\lambda I[/itex]?

Now, use that

[tex]A^2-\lambda I=(A-\sqrt{\lambda}I)(A+\sqrt{\lambda} I)[/tex]
 
  • #3
micromass said:
Let [itex]\lambda[/itex] be an eigenvalue of [itex]A^2[/itex]. What can you say about [itex]A^2-\lambda I[/itex]?

Now, use that

[tex]A^2-\lambda I=(A-\sqrt{\lambda}I)(A+\sqrt{\lambda} I)[/tex]

Then:

[tex]det(A-\sqrt{\lambda}I)*det(A+\sqrt{\lambda} I) = 0[/tex]

Not sure how to proceed...
 
  • #4
If xy=0, then what can you say about x or y?
 
  • #5
micromass said:
If xy=0, then what can you say about x or y?

Oh so all we can say is that either [itex]\sqrt{\lambda}[/itex] or [itex]-\sqrt{\lambda}[/itex] is an eigenvalue of A, but not both?
 
  • #6
IniquiTrance said:
Oh so all we can say is that either [itex]\sqrt{\lambda}[/itex] or [itex]-\sqrt{\lambda}[/itex] is an eigenvalue of A, but not both?

Right.
 
  • #7
Thank you.
 
  • #8
One more question, how do we know that all the eigenvalues of [itex]A^2[/itex] are positive? Since A is symmetric, if [itex]\lambda < 0[/itex] were an eigenevalue of [itex]A^2[/itex] we'd run into a problem...
 
  • #9
We can do the same with [itex]\sqrt{\lambda}[/itex] complex numbers, no?
 
  • #10
micromass said:
We can do the same with [itex]\sqrt{\lambda}[/itex] complex numbers, no?

But since A is symmetric, it must have real eigenvalues, no?
 
  • #11
IniquiTrance said:
But since A is symmetric, it must have real eigenvalues, no?

Yes, sure. So what we do is we argue the same argument where [itex]\lambda[/itex] might be negative and where [itex]\sqrt{\lambda}[/itex] might be complex. Then we can deduce that either [itex]\sqrt{\lambda}[/itex] or [itex]-\sqrt{\lambda}[/itex] are eigenvalues of A. Since A is symmetric, we have that [itex]\sqrt{\lambda}[/itex] must be real, so [itex]\lambda[/itex] was positive to begin with.
 
  • #12
micromass said:
Yes, sure. So what we do is we argue the same argument where [itex]\lambda[/itex] might be negative and where [itex]\sqrt{\lambda}[/itex] might be complex. Then we can deduce that either [itex]\sqrt{\lambda}[/itex] or [itex]-\sqrt{\lambda}[/itex] are eigenvalues of A. Since A is symmetric, we have that [itex]\sqrt{\lambda}[/itex] must be real, so [itex]\lambda[/itex] was positive to begin with.

Ah ok, thanks again!
 
  • #13
IniquiTrance said:
Ah ok, thanks again!

No, you can have both. The matrix
[tex] A = \pmatrix{2 & 0 \\0 & -2}[/tex] has
[tex] B = A^2 = \pmatrix{4 & 0 \\0 & 4}, [/tex]
and so both [itex] \sqrt{4}[/itex] and [itex] -\sqrt{4}[/itex] are eigenvalues of [itex]A.[/itex]

RGV
 

Related to Eigenvalues of the power of a matrix

1. What are eigenvalues of the power of a matrix?

The eigenvalues of the power of a matrix refer to the eigenvalues of the matrix raised to a certain power. The eigenvalues are the values that satisfy the equation Ax = λx, where A is the matrix and x is the eigenvector.

2. How do eigenvalues of the power of a matrix relate to the original matrix?

The eigenvalues of the power of a matrix are related to the original matrix by the fact that the eigenvalues of the power of a matrix are the eigenvalues of the original matrix raised to the same power. In other words, if A is the original matrix and λ is an eigenvalue of A, then λ^n is an eigenvalue of A^n.

3. Can the eigenvalues of the power of a matrix be negative?

Yes, the eigenvalues of the power of a matrix can be negative. The eigenvalues of a matrix can be any complex number, including negative numbers. However, the eigenvalues of a real matrix (a matrix with real numbers) will always have a real component.

4. How are the eigenvalues of the power of a matrix calculated?

The eigenvalues of the power of a matrix can be calculated by first finding the eigenvalues of the original matrix. Then, raise each eigenvalue to the desired power to get the eigenvalues of the power of the matrix.

5. What is the significance of eigenvalues of the power of a matrix?

The eigenvalues of the power of a matrix have many applications in mathematics and science. They are used in solving systems of linear equations, analyzing the stability of dynamical systems, and finding the dominant behavior of a matrix. They also have applications in fields such as physics, engineering, and economics.

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