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Let $\Sigma$ be a covariance matrix. According to the material in this link,

If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative

this is actually written about the correlation matrix, but the principle should be the same.

What does "most" here mean? Is there a common condition that would make all the off-diagonal elements negative?

Let $\Sigma$ be a covariance matrix. According to the material in this link,

If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative.

This is actually written about the correlation matrix, but the principle should be the same.

What does "most" here mean? Is there a common condition that would make all the off-diagonal elements negative?

Let $\Sigma$ be a covariance matrix. According to link,

If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative

this is actually written about the correlation matrix, but the principle should be the same.

What does "most" here mean? Is there a common condition that would make all the off-diagonal elements negative?

Let $\Sigma$ be a covariance matrix. According to the material in this link,

If the elements of $\Sigma$ are all positive, most of the off-diagonal elements in $\Sigma^{-1}$ will be negative

this is actually written about the correlation matrix, but the principle should be the same.

What does "most" here mean? Is there a common condition that would make all the off-diagonal elements negative?