How Accurate Are the Bounds for Eigenvalues in Circulant Matrices?

[EngWiPy]

Hi,

I have the following equation:

$$\gamma=\frac{1}{\frac{1}{N}\sum_{n=1}^N|\lambda_n|^{-2}}$$

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

I used two properties to bound the above equation:

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

$$\sum_{n=1}^N|\lambda_n|^{-2}\leq\left(\sum_{n=1}^N|\lambda_n|^{2}\right)^{-1}$$

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:

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

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.