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Added tag. Minor edits.
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edit approved Jun 28, 2020 at 23:10
Rodrigo de Azevedo
edit approved Jun 28, 2020 at 23:10
Rodrigo de Azevedo
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Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}). $$

$$ \text{Tr} \left( G^2 \right) \leq \text{Tr} \left( GUGU^{-1} \right). $$

If $U$ and $G$ commute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to   

$$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and

and then using the Cauchy-Schwarz inequality. Unfortunately, I have not found a solution yet.

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}). $$

If $U$ and $G$ commute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to  $$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and then using the Cauchy-Schwarz inequality. Unfortunately, I have not found a solution yet.

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality

$$ \text{Tr} \left( G^2 \right) \leq \text{Tr} \left( GUGU^{-1} \right). $$

If $U$ and $G$ commute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to 

$$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$

and then using the Cauchy-Schwarz inequality. Unfortunately, I have not found a solution yet.

some clarity / formatting
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Federico Poloni
Federico Poloni
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Let $G \in \mathbb{R}^{3x3}$$G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3x3}$$U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. Now,I would like to prove the following inequality shall be proved: $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}) $$ If$$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}). $$

If $U$ and $G$ have the same eigenvectorscommute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to $$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and then using the Cauchy-Schwarz inequality in order to achieve an upper boundary that possibly fulfils the inequality. Unfortunately, I have not found a solution yet.

Thanks for any advice!

Let $G \in \mathbb{R}^{3x3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3x3}$ a symmetric and positive definite matrix. Now, the following inequality shall be proved: $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}) $$ If $U$ and $G$ have the same eigenvectors, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to $$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and then using the Cauchy-Schwarz inequality in order to achieve an upper boundary that possibly fulfils the inequality. Unfortunately, I have not found a solution yet.

Thanks for any advice!

Let $G \in \mathbb{R}^{3 \times 3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3\times 3}$ a symmetric and positive definite matrix. I would like to prove the inequality $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}). $$

If $U$ and $G$ commute, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to $$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and then using the Cauchy-Schwarz inequality. Unfortunately, I have not found a solution yet.

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TARS
TARS
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Trace inequality under consideration of definiteness

Let $G \in \mathbb{R}^{3x3}$ a symmetric, but indefinite matrix and $U \in \mathbb{R}^{3x3}$ a symmetric and positive definite matrix. Now, the following inequality shall be proved: $$ \text{Tr}(G^2) \leq \text{Tr}(GUGU^{-1}) $$ If $U$ and $G$ have the same eigenvectors, both sides of the inequality are obviously equal. However for more general cases, I have tried to rearrange the inequality to $$ \text{Tr}(\underbrace{[UG-GU]}_{\text{skew-symmetric}}\ GU^{-1}) \leq 0 $$ and then using the Cauchy-Schwarz inequality in order to achieve an upper boundary that possibly fulfils the inequality. Unfortunately, I have not found a solution yet.

Thanks for any advice!