What is the distribution of engenvalues of covariance matrix when the covariance has some block diagonal structure

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Let's say we have a matrix $X inmathbb R^ntimes p$, where $X_i,j$ sampled from a Gaussian $N(mu, sigma^2)$, we use $Phi$ to denote $mu,sigma$ for simplicity.

Now, we sample $m$ different Matrices $X_k$, parametrized by $Phi_k$ respectively, where $k = 1,dots m$.

Append them to make a matrix $Ain R^mntimes p$, calculate covariance $C=AA^T$. When we visualize $C$, we should clearly see the block-diagonal structure.

When we calculate and plot the eigenvalues, we can (mostly) clearly see that there are $m$ significant ones.

Now, do we know anything about the distribution of these eigenvalues (as a function of $Phi$)?

edited Jul 26 at 22:29

Michael Hardy

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asked Jul 26 at 20:51

Haohan Wang

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up vote
1
down vote

favorite

Let's say we have a matrix $X inmathbb R^ntimes p$, where $X_i,j$ sampled from a Gaussian $N(mu, sigma^2)$, we use $Phi$ to denote $mu,sigma$ for simplicity.

Now, we sample $m$ different Matrices $X_k$, parametrized by $Phi_k$ respectively, where $k = 1,dots m$.

Append them to make a matrix $Ain R^mntimes p$, calculate covariance $C=AA^T$. When we visualize $C$, we should clearly see the block-diagonal structure.

When we calculate and plot the eigenvalues, we can (mostly) clearly see that there are $m$ significant ones.

Now, do we know anything about the distribution of these eigenvalues (as a function of $Phi$)?

edited Jul 26 at 22:29

Michael Hardy

204k23186461

asked Jul 26 at 20:51

Haohan Wang

213110

add a commentÂ |Â

up vote
1
down vote

favorite

Let's say we have a matrix $X inmathbb R^ntimes p$, where $X_i,j$ sampled from a Gaussian $N(mu, sigma^2)$, we use $Phi$ to denote $mu,sigma$ for simplicity.

Now, we sample $m$ different Matrices $X_k$, parametrized by $Phi_k$ respectively, where $k = 1,dots m$.

Append them to make a matrix $Ain R^mntimes p$, calculate covariance $C=AA^T$. When we visualize $C$, we should clearly see the block-diagonal structure.

When we calculate and plot the eigenvalues, we can (mostly) clearly see that there are $m$ significant ones.

Now, do we know anything about the distribution of these eigenvalues (as a function of $Phi$)?

edited Jul 26 at 22:29

Michael Hardy

204k23186461

asked Jul 26 at 20:51

Haohan Wang

213110

Let's say we have a matrix $X inmathbb R^ntimes p$, where $X_i,j$ sampled from a Gaussian $N(mu, sigma^2)$, we use $Phi$ to denote $mu,sigma$ for simplicity.

Now, we sample $m$ different Matrices $X_k$, parametrized by $Phi_k$ respectively, where $k = 1,dots m$.

Append them to make a matrix $Ain R^mntimes p$, calculate covariance $C=AA^T$. When we visualize $C$, we should clearly see the block-diagonal structure.

When we calculate and plot the eigenvalues, we can (mostly) clearly see that there are $m$ significant ones.

Now, do we know anything about the distribution of these eigenvalues (as a function of $Phi$)?

edited Jul 26 at 22:29

Michael Hardy

204k23186461

asked Jul 26 at 20:51

Haohan Wang

213110

edited Jul 26 at 22:29

Michael Hardy

204k23186461

edited Jul 26 at 22:29

Michael Hardy

204k23186461

edited Jul 26 at 22:29

Michael Hardy

204k23186461

asked Jul 26 at 20:51

Haohan Wang

213110

asked Jul 26 at 20:51

Haohan Wang

213110

asked Jul 26 at 20:51

Haohan Wang

213110

add a commentÂ |Â

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