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I am reading a paper where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such as $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \left\vert B^\top \Sigma^{-1} B \right\vert \leq \log \det \left\vert B^\top \Delta^{-1} B \right\vert.$$

Could someone point me in the direction of explaining why is it true?

I am thinking of using Poincaré separation theorem, which provides the bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \text{Var}(X)$, and $\Delta = E (\text{Var}(X|y))$, so $\Sigma-\Delta = \text{Var}(E(X \vert y))$ is also positive definite. The $B$ in my question plays the role as $𝐵_0$ in the paper.

I am reading a paper Cook and Forzani - Likelihood-Based Sufficient Dimension Reduction where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such that $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \left\lvert B^\top \Sigma^{-1} B \right\rvert \leq \log \det \left\lvert B^\top \Delta^{-1} B \right\rvert.$$

Could someone point me in the direction of explaining why it is true?

I am thinking of using the Poincaré separation theorem, which provides bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \operatorname{Var}(X)$, and $\Delta = E (\operatorname{Var}(X\vert y))$, so $\Sigma-\Delta = \operatorname{Var}(E(X \vert y))$ is also positive definite. The $B$ in my question plays the role of $B_0$ in the paper.

I am reading a paper where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such as $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \vert B^\top \Sigma^{-1} B \vert \leq \log \det \vert B^\top \Delta^{-1} B \vert.$$

Could someone point me in the direction of explaining why is it true?

I am thinking of using Poincaré separation theorem, which provides the bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \text{Var}(X)$, and $\Delta = E (\text{Var}(X|y))$, so $\Sigma-\Delta = \text{Var}(E(X \vert y))$ is also positive definite. The B in my question plays the role as $𝐵_0$ in the paper.

I am reading a paper where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such as $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \left\vert B^\top \Sigma^{-1} B \right\vert \leq \log \det \left\vert B^\top \Delta^{-1} B \right\vert.$$

Could someone point me in the direction of explaining why is it true?

I am thinking of using Poincaré separation theorem, which provides the bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \text{Var}(X)$, and $\Delta = E (\text{Var}(X|y))$, so $\Sigma-\Delta = \text{Var}(E(X \vert y))$ is also positive definite. The $B$ in my question plays the role as $𝐵_0$ in the paper.

I am reading a paper where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such as $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \vert B^\top \Sigma^{-1} B \vert \leq \log \det \vert B^\top \Delta^{-1} B \vert.$$

Could someone point me in the direction of explaining why is it true?

I am thinking of using Poincaré separation theorem, which provides the bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

Edit: The paper is included for the context of the question. The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \text{Var}(X)$, and $\Delta = E (\text{Var}(X|y))$, so $\Sigma-\Delta = \text{Var}(E(X \vert y))$ is also positive definite. The B in my question plays the role as $𝐵_0$ in the paper.

I am reading a paper where the author uses the following result from matrix analysis but does not explain why it is true nor provide any reference.

More specifically, let $B \in \mathbb{R}^{p\times d}$ be a semi-orthogonal matrix, i.e $B^\top B = I_d$, and $d < p$. Let $\Sigma$ and $\Delta$ denote two symmetric positive definite matrices such as $\Sigma - \Delta$ is also positive definite. What they claim is that

$$ \log \det \vert B^\top \Sigma^{-1} B \vert \leq \log \det \vert B^\top \Delta^{-1} B \vert.$$

Could someone point me in the direction of explaining why is it true?

I am thinking of using Poincaré separation theorem, which provides the bounds on the eigenvalues of the matrices on both the left and right-hand sides; however, I did not make any progress with it.

The claim is in the proof of Proposition 3, at the top of page 33. Here $\Sigma = \text{Var}(X)$, and $\Delta = E (\text{Var}(X|y))$, so $\Sigma-\Delta = \text{Var}(E(X \vert y))$ is also positive definite. The B in my question plays the role as $𝐵_0$ in the paper.