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.










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  • 1




    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
    – Eric
    Nov 27 '18 at 10:34


















0














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.










share|cite|improve this question


















  • 1




    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
    – Eric
    Nov 27 '18 at 10:34
















0












0








0







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.










share|cite|improve this question













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






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share|cite|improve this question











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share|cite|improve this question










asked Nov 27 '18 at 9:50









ShewShew

563413




563413








  • 1




    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
    – Eric
    Nov 27 '18 at 10:34
















  • 1




    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
    – Eric
    Nov 27 '18 at 10:34










1




1




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
– Eric
Nov 27 '18 at 10:34






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
– Eric
Nov 27 '18 at 10:34












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