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Peter
Peter
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Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deterministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and, $n$ and $m=o(\sqrt{n})$, this is known to be Wishart, $W_P(T^HT,n)$$W_P(T^HT,m)$. But I'm interested in the answer for small values of $m$ and $n$.

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deterministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and $n$ this is known to be Wishart, $W_P(T^HT,n)$. But I'm interested in the answer for small values of $m$ and $n$.

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deterministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$, $n$ and $m=o(\sqrt{n})$, this is known to be Wishart, $W_P(T^HT,m)$. But I'm interested in the answer for small values of $m$ and $n$.

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Peter
Peter
  • 141
  • 3

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{c}^{n\times m}$$\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deternministicdeterministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and $n$ this is known to be Wishart, $W_P(T^HT,n)$. But I'm interested in the answer for small values of $m$ and $n$.

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{c}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deternministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and $n$ this is known to be Wishart, $W_P(T^HT,n)$. But I'm interested in the answer for small values of $m$ and $n$.

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deterministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and $n$ this is known to be Wishart, $W_P(T^HT,n)$. But I'm interested in the answer for small values of $m$ and $n$.

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Peter
Peter
  • 141
  • 3

Distribution of random Projections on the Stiefel Manifold

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{c}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar Measure. Also assume $\mathbf{T}\in\mathbb{C}^{n\times p}$, $p<m<n$ is a deternministic and known matrix. I wonder what the distribution of $\mathbf{T}^H\mathbf{U}\mathbf{U}^H\mathbf{T}$ is? For large $m$ and $n$ this is known to be Wishart, $W_P(T^HT,n)$. But I'm interested in the answer for small values of $m$ and $n$.