How Accurate Are the Bounds for Eigenvalues in Circulant Matrices?

[EngWiPy]

Hi,

I have the following equation:

[tex]\gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}}[/tex]

where lambdas are the eigenvalues of an N-by-N circulant matrix A.

I used two properties to bound the above equation:

[tex]\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}\geq\left(\prod_{n=1}^N|\lambda_n|^{-2}\right)^{1/N}[/tex]

[tex]\sum_{n=1}^N|\lambda_n|^{-2}\leq\left(\sum_{n=1}^N|\lambda_n|^{2}\right)^{-1}[/tex]

Are these two bounds correct?

Thanks

 

[mathman]

The first one is a a statement that the arithmetic mean bounds the geometric mean (true).

The second is false - simple example: 2 terms on the left 1 + 2 = 3. Right side is 1/(1+1/2) = 2/3.

 

[EngWiPy]

The first one is a a statement that the arithmetic mean bounds the geometric mean (true).

The second is false - simple example: 2 terms on the left 1 + 2 = 3. Right side is 1/(1+1/2) = 2/3.

But we have the identity for the trace that:

[tex]\sum_{n=1}^N\lambda_n^k=\text{Tr}\left(\mathbf{A}^k\right)\leq\left[\text{Tr}\left(\mathbf{A}\right)\right]^k=\left(\sum_{n=1}^N\lambda_n\right)^k[/tex]

or it just works for k>=1?

 

[mathman]

I am not familiar with the trace equation, but as my example shows, what you propose is just wrong. It may be that your guess is correct, k is positive integer.

 

[CellCoree]

for your question. It appears that the bounds you have used are correct based on the properties of arithmetic and geometric mean. The first bound uses the property that the arithmetic mean is always greater than or equal to the geometric mean, while the second bound uses the property that the sum of squares is always greater than or equal to the square of the sum. These properties can be applied to the equation you have provided to bound the value of \gamma. However, it is always important to double check your calculations and make sure they are appropriate for the specific problem at hand.