Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to  ?

Note. I'm really only interested in (good) lower-bounds.

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1}\label{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\lVert x\rVert = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in \eqref{1} by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to?

Note. I'm really only interested in (good) lower-bounds.

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \operatorname{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_\text{max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\lVert P_V u\rVert^2 = (k/n)\lVert u\rVert^2$.

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to ?

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to ?

Note. I'm really only interested in (good) lower-bounds.

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$ eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to ?

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant-Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by

$$ \tag{1} \lambda_k = \min_{V \in G_{n-k+1,n}} R(A,V), $$ where $R(A,V):= \max_{x \in V,\,\|x\| = 1} x^\top A x$.

Now, let $V$ be drawn according to the Haar distribution on $G_{k,n}$, and replace the min in (1) by expectation over $V$.

Question. What does $\alpha_k(A) := \mathbb E_V [R(A,V)]$ correspond / evaluate to ?

Examples

Let $P_V$ be the orthogonal projector for $V$.

• If $k=1$, then $R(A,V) = v^\top A v$, where $v$ is uniform on the unit-sphere in $\mathbb R^n$, and so $\alpha_1(A) = \mbox{trace}(A)/n$.
• If $k=n$, then $P_V = I_n$ with probability $1$ and so $\alpha_n(A) = \lambda_{\max}(A)$.
• If $A = I_n$, then obviously $\alpha_k(A) = 1$ for all $k \in [n]$.
• If $A = uu^\top$, a rank 1 matrix, then $\alpha_k(A) = \mathbb E_V R(A,V) = \mathbb E_V\|P_V u\|^2 = (k/n)\|u\|^2$.