Condition for a symmetric matrix to contain only positive entries?
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I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
ShewShew
570413
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add a comment |
$begingroup$
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
ShewShew
570413
$endgroup$
1
$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34
add a comment |
$begingroup$
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
ShewShew
570413
$endgroup$
I have a matrix defined as $mathbf{X} -mathbf{Y}mathbf{Y}^top$, $mathbf{Y}$ is a matrix containing only non-negative entries and $mathbf{X}$ is a similarity matrix, hence $mathbf{X}$ also contains only non-negative entries. The matrix $mathbf{X}$ is a PSD matrix and $mathbf{X} -mathbf{Y}mathbf{Y}^top$ is a symmetric matrix.
Now, is there any condition (other than the trivial condition that element-wise $mathbf{X} geq mathbf{YY}^top$).
I am looking for any condition in terms of the spectral properties of the matrices or the bound on the supremum of elements of the matrices etc.
linear-algebra matrices matrix-decomposition positive-semidefinite
linear-algebra matrices matrix-decomposition positive-semidefinite
asked Nov 27 '18 at 9:50
ShewShew
570413
asked Nov 27 '18 at 9:50
ShewShew
570413
asked Nov 27 '18 at 9:50
ShewShew
570413
asked Nov 27 '18 at 9:50
ShewShew
570413
asked Nov 27 '18 at 9:50
ShewShew
570413
570413
1
$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34
add a comment |
1
$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34
1
1
$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34
$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34
add a comment |
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$begingroup$
Matrices of the form $YY^T$ where $Y$ has nonnegative entries are called completely positive matrices. There is quite an extensive study of them if you google the term. I haven't read the following paper, but it claims to provide a necessary and sufficient condition for a matrix to be CP. sciencedirect.com/science/article/pii/S0024379503006116. Beyond that, while I am not an expert on CP matrices, from the half week I spent on them I recall that the classification of CP matrices is not a simple problem and for example, checking the completely positive rank of a matrix is NP-hard
$endgroup$
– Eric
Nov 27 '18 at 10:34