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Fixed from addition to subtraction of terms in second equation.
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haz
haz
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Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.

Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ denote the vector of all ones.

I'm using the matrix inversion lemma (Woodbury formula) to rewrite $${\bf y'} = f({\bf A'})^{-1}f({\bf A}){\bf y}$$ as $${\bf y'} = {\bf y} + f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$$${\bf y'} = {\bf y} - f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$ where I use the eigendecomposition $${\bf P = Q^{-1}\Lambda Q}.$$

Unfortunately, ${\bf A'}$ is unknown, so ${\bf P}$ is a random symmetric matrix that is constrained away from being pure Bernoulli. (Even if it were Bernoulli, it appears most of the random matrix literature focuses on the case of p=0.5, whereas I'm most interested in the case of p << 0.5.)

Is it possible to determine the probability distribution of ${\bf Q}$ and ${\bf \Lambda}$ at this point in the random matrix literature? (I am an outsider in an applied field.)

Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.

Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ denote the vector of all ones.

I'm using the matrix inversion lemma (Woodbury formula) to rewrite $${\bf y'} = f({\bf A'})^{-1}f({\bf A}){\bf y}$$ as $${\bf y'} = {\bf y} + f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$ where I use the eigendecomposition $${\bf P = Q^{-1}\Lambda Q}.$$

Unfortunately, ${\bf A'}$ is unknown, so ${\bf P}$ is a random symmetric matrix that is constrained away from being pure Bernoulli. (Even if it were Bernoulli, it appears most of the random matrix literature focuses on the case of p=0.5, whereas I'm most interested in the case of p << 0.5.)

Is it possible to determine the probability distribution of ${\bf Q}$ and ${\bf \Lambda}$ at this point in the random matrix literature? (I am an outsider in an applied field.)

Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.

Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ denote the vector of all ones.

I'm using the matrix inversion lemma (Woodbury formula) to rewrite $${\bf y'} = f({\bf A'})^{-1}f({\bf A}){\bf y}$$ as $${\bf y'} = {\bf y} - f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$ where I use the eigendecomposition $${\bf P = Q^{-1}\Lambda Q}.$$

Unfortunately, ${\bf A'}$ is unknown, so ${\bf P}$ is a random symmetric matrix that is constrained away from being pure Bernoulli. (Even if it were Bernoulli, it appears most of the random matrix literature focuses on the case of p=0.5, whereas I'm most interested in the case of p << 0.5.)

Is it possible to determine the probability distribution of ${\bf Q}$ and ${\bf \Lambda}$ at this point in the random matrix literature? (I am an outsider in an applied field.)

Source Link
haz
haz
  • 43
  • 6

Distribution of eigenvectors and eigenvalues for random, symmetric matrix

Consider two simple, undirected graphs with adjacency matrices ${\bf A}$ and ${\bf A'}$. Let ${\bf P} = {\bf A'} - {\bf A}$. Thus, ${\bf P}$ is symmetric and always 0 along the diagonal.

Let $f({\bf X})$ be a (undisclosed) linear transformation of a matrix $X$. Let ${\bf y}$ and ${\bf y'}$ be vectors, and ${\bf 1}$ denote the vector of all ones.

I'm using the matrix inversion lemma (Woodbury formula) to rewrite $${\bf y'} = f({\bf A'})^{-1}f({\bf A}){\bf y}$$ as $${\bf y'} = {\bf y} + f({\bf A})^{-1}{\bf Q}\left[{\bf \Lambda^{-1}} + {\bf Q}f({\bf A})^{-1}{\bf Q}\right]^{-1}{\bf Q}^{-1}{\bf y}$$ where I use the eigendecomposition $${\bf P = Q^{-1}\Lambda Q}.$$

Unfortunately, ${\bf A'}$ is unknown, so ${\bf P}$ is a random symmetric matrix that is constrained away from being pure Bernoulli. (Even if it were Bernoulli, it appears most of the random matrix literature focuses on the case of p=0.5, whereas I'm most interested in the case of p << 0.5.)

Is it possible to determine the probability distribution of ${\bf Q}$ and ${\bf \Lambda}$ at this point in the random matrix literature? (I am an outsider in an applied field.)