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Real Geometry of Matrix Completion
Rainer Sinn (Freie Universität Berlin)
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Symmetric matrix completion
����� ? −�? � ?−� ? −�
����Constraints:(�) Rank (smallest possible, fixed, …)(�) Positive semidefinite
����∗ ? ∗? ∗ ?∗ ? ∗
����Spectral Theorem
Definition. A real symmetric matrix is positive semidefinite if all of its real eigen-values are nonnegative.
Theorem. A real symmetricmatrixM is positive semidefinite if andonly if there existsa real matrix B such thatM = BTB.
Real symmetric matricesM are in one-to-one correspondence with real quadraticforms QM = (x�, x�, . . . , xn)M(x�, x�, . . . , xn)T .Example.
M =������
� � −� �� � � −��−� � � ��� −�� �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� − �x�x� + �x�x� + ��x�x� − ��x�x� + ��x�x�
�
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
Nonnegative Quadratic Forms and Sums of Squares
Theorem. A quadratic form QM = xTMx is positive semidefinite if and only if it is asum of squaresQM = (Bx)T(Bx).Example.
��x�� +���x��+���x��+���x�x�+���x�x�+���x�x� = (�x�−x�+�x�)�+(�x�+��x�−��x�)�
������ �� ����� ��� ������ ��� ���
���� =����� �−� ��� −��
������ −� �� �� −���
Smallest number of squares in QM = ℓ�� + . . . + ℓ�r = rank ofM
Obvious Necessary Condition
Theorem (Sylvester’s criterion). A symmetric matrix is positive semidefinite if andonly if all of its principal minors are nonnegative.
Example.
M = ����� � ?� � ?? ? �
����QM = x�� + x�� + x�� + �x�x�+?x�x�+?x�x�
QM(�,−�, �) = � + � − � = −�Example.
������
� � ? −�� � � ?? � � �−� ? � �
������
Aside: diagonal entries
����� ? ?? � �? � ?
����
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
References
Thank you
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
References
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����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
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Positive Semidefinite Matrix Completion
Let G = ([n], E) be a simple graph on n vertices.
Definition. A G-partial matrix is the image of a symmetric matrix under the pro-jection
πG∶� Symn(R)→ Rn ⊕R�E
A = (ai j)� ((a��, . . . , ann), (ai j∶{i , j} ∈ E))Example.
������
a�� a�� a�� ∗a�� a�� a�� ∗a�� a�� a�� a��∗ ∗ a�� a��
������∗ ∗ ∗ ?∗ ∗ ∗ ?∗ ∗ ∗ ∗? ? ∗ ∗
��������
��������Example.
������
a�� a�� ∗ a��a�� a�� a�� ∗∗ a�� a�� a��a�� ∗ a�� a��
�������
� � �
∗ ∗ ? ∗∗ ∗ ∗ ?? ∗ ∗ ∗∗ ? ∗ ∗
��������
���������
� �
�
Question. Given a G-partial matrix, does there exist a positive semidefinite sym-metric completion?
�
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
References
Thank you
����� ? ?? � �? � ?
����Patterns and graphs
Complete principal minors: Cliques
Stanley-Reisner Ideals
Definition. The Stanley-Reisner ideal associated to G = ([n], E) is the square-free monomial ideal
IG = �xix j∶ {i , j} ∉ E� ⊂ R[x�, x�, . . . , xn].Example.
IG = �x�x�, x�x��
M =������
� � ? �� � � ?? � � ��� ? �� �
������QM = x�� + x�� + x�� + x�� + ��x�x� + �x�x� + ��x�x� + ��x�x�+?x�x�+?x�x�
Proposition. (�)G-partialmatricesM are inone-to-onecorrespondencewith residueclasses of quadratic formsQM inR[x�, x�, . . . , xn]�IG = R[XG].(�) AG-partial matrixM has a positive semidefinite completion if and only if the cor-responding quadratic formQM is a sum of squares of linear formsmodulo IG.
Minimal Free Resolutions
Let S = R[x�, x�, . . . , xn].Definition. A minimal free resolution of a graded S-module M is an exact se-quence
� δt+��→ Ftδt�→ Ft−� δt−��→ � δ��→ F� → M → �,
where the Fi are free S-modules and the image of the map δi is contained in thesubmodule (x�, x�, . . . , xn)Fi−� of Fi−� for all i ≥ �.Example. ConsiderM� = S��x�x�, x�x�� andM� = S��x�x�, x�x��.
�→ S(x�,−x�)T����→ S�
(x�x�,x�x�)�����→ S → M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ S → M� → �
areminimal free resolutions ofM� andM�.
Definition. A graded S-module M is �-regular (Castelnuovo-Mumford) if everythe degree of every entry of the matrices representing δi is �.
Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
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Example. ConsiderM� = �x�x�, x�x�� andM� = �x�x�, x�x��.�→ S
(x�,−x�)T����→ S�(x�x�,x�x�)�����→ M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ M� → �
areminimal free resolutions ofM� andM�.
Definition. A graded S-moduleM is �-regular (Castelnuovo-Mumford) if the en-tries of δ� have degree � and the entries of δi have degree � for all i > �.Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
The Hankel Spectrahedron
Definition. The convex dual cone of ΣG is the Hankel spectrahedron of G.
Proposition.
Σ∨G = S+ ∩ (ker(πG))⊥Proof.
Σ∨G = (πG(S+))∨ = S+ ∩ (ker(πG))⊥
Example. ConsiderM� = �x�x�, x�x�� andM� = �x�x�, x�x��.�→ S
(x�,−x�)T����→ S�(x�x�,x�x�)�����→ M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ M� → �
areminimal free resolutions ofM� andM�.
Definition. A homogeneous ideal I ⊂ S is �-regular (Castelnuovo-Mumford) if itis generated by quadrics and the entries of δi have degree � for all i ≥ �.Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
The Hankel Spectrahedron
Definition. The convex dual cone of ΣG is the Hankel spectrahedron of G.
Proposition.
Σ∨G = S+ ∩ (ker(πG))⊥Proof.
Σ∨G = (πG(S+))∨ = S+ ∩ (ker(πG))⊥
Example. ConsiderM� = S��x�x�, x�x�� andM� = S��x�x�, x�x��.�→ S
(x�,−x�)T����→ S�(x�x�,x�x�)�����→ S → M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ S → M� → �
areminimal free resolutions ofM� andM�.
Definition. A graded S-module M is �-regular (Castelnuovo-Mumford) if everythe degree of every entry of the matrices representing δi is �.
Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
The Hankel Spectrahedron
Definition. The convex dual cone of ΣG is the Hankel spectrahedron of G.
Proposition.
Σ∨G = S+ ∩ (ker(πG))⊥Proof.
Σ∨G = (πG(S+))∨ = S+ ∩ (ker(πG))⊥
Example. ConsiderM� = S��x�x�, x�x�� andM� = S��x�x�, x�x��.�→ S
(x�,−x�)T����→ S�(x�x�,x�x�)�����→ S → M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ S → M� → �
areminimal free resolutions ofM� andM�.
Definition. A graded S-module M is �-regular (Castelnuovo-Mumford) if everythe degree of every entry of the matrices representing δi is �.
Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
The Hankel Spectrahedron
Definition. The convex dual cone of ΣG is the Hankel spectrahedron of G.
Proposition.
Σ∨G = S+ ∩ (ker(πG))⊥Proof.
Σ∨G = (πG(S+))∨ = S+ ∩ (ker(πG))⊥
Example. ConsiderM� = S��x�x�, x�x�� andM� = S��x�x�, x�x��.�→ S
(x�,−x�)T����→ S�(x�x�,x�x�)�����→ S → M� → �
�→ S(x�x�,−x�x�)T������→ S�
(x�x�,x�x�)�����→ S → M� → �
areminimal free resolutions ofM� andM�.
Definition. A graded S-module M is �-regular (Castelnuovo-Mumford) if everythe degree of every entry of the matrices representing δi is �.
Theorem (Fröberg). The ideal IG is �-regular if and only ifG is chordal.
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
The Hankel Spectrahedron
Definition. The convex dual cone of ΣG is the Hankel spectrahedron of G.
Proposition.
Σ∨G = S+ ∩ (ker(πG))⊥Proof.
Σ∨G = (πG(S+))∨ = S+ ∩ (ker(πG))⊥Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Hankel index and bpf linear series
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
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Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Hankel index and bpf linear series
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
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Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Hankel index and bpf linear series
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Extreme Rays of the Dual Convex Cone
Proposition. LetG = ([m], E) be them-cycle. Thematrix
����������������
m−�m−� −� � � . . . � � �
m−�−� � −� � . . . � � �� −� � −� . . . ⋮ ⋮ ⋮⋮ � −� � � ⋮ ⋮ ⋮⋮ ⋮ � � � ⋮� � � � . . . � −� �� � � � . . . −� � −��
m−� � � � . . . � −� m−�m−�
����������������is an extreme ray of Σ∨G. It has rankm − �. It certifies that
�����������
� � ? ? . . . ? −�� � � ? . . . ? ?? � � � . . . ? ?⋮ � � � ⋮? ? ? ? . . . � �−� ? ? ? . . . � �
�����������does not have a positive semidefinite completion even though every complete princi-pal minor is positive semidefinite.
References
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Theorem (Blekherman, Sinn).
gcr(Km,m) = mmlt(Km,m) =min�k∶ �k + �
�� ≥ �m� ∈ O(�√m).
GramDimension (SOS length)
Definition (Laurent,Varvitsiotis). TheGramdimensionof a labeled simplegraphG = ([n], E) is the smallest whole number k such that for any positive semidef-inite matrix M ∈ S+, there exists another M′ ∈ S+ of rank at most k such thatπG(M) = πG(M′).Theorem (Laurent, Varvitsiotis). (�) k ≤ �: The Gramdimension ofG is atmost k ifand only ifG has no Kk+�minor.(�) The Gram dimension ofG is at most � if and only ifG has no K� and K�,�,�minors.
Example. The Gram dimension of anym-cylce is �.
m = �References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
References
Thank you
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
Positive Definite Completions
Theorem. Letm the smallest length of a chordless cycle in G. A G-partial matrixMhas a positive definite completion if and only if(�) every complete principal minor ofM is positive, and(�)M has a positive semidefinite completion of rank at least n −m + �.
MLT vs. GCR
Let G = ([n], E) be a labeled simple graph.
Definition. (�) The maximum likelihood threshold of G is the smallest wholenumber k such that there exists a positive definite matrix D with πG(M) = πG(D)for almost all positive semidefinite matricesM of rank at most k.(�) The generic completion rank of G is the smallest whole number k such thatdim(πG(Vk)) = n + �E, where Vk is the variety of matrices of rank at most k.
Proposition (Sard’sTheoremand Implicit FunctionTheorem). gcr(G) ≥mlt(G)
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
Positive Definite Completions
Theorem. Letm the smallest length of a chordless cycle in G. A G-partial matrixMhas a positive definite completion if and only if(�) every complete principal minor ofM is positive, and(�)M has a positive semidefinite completion of rank at least n −m + �.
MLT vs. GCR
Let G = ([n], E) be a labeled simple graph.
Definition. (�) The maximum likelihood threshold of G is the smallest wholenumber k such that there exists a positive definite matrix D with πG(M) = πG(D)for almost all positive semidefinite matricesM of rank at most k.(�) The generic completion rank of G is the smallest whole number k such thatdim(πG(Vk)) = n + �E, where Vk is the variety of matrices of rank at most k.
Proposition (Sard’sTheoremand Implicit FunctionTheorem). gcr(G) ≥mlt(G)
Question:What are the extreme rays of Σ∨G?Proposition. Extreme rays of Σ∨G that have rank � are in one-to-one correspondencewith points x ∈ V(IG) = XG.
Proof. R = xxT ∈ Σ∨G,M ∈ ker(πG)) ≅ IG:� = �M , R� = tr(MR) = tr(MxxT) = xTMx = QM(x)
Hankel index and bpf linear series
Definition. The Hankel index of G is the smallest r > � such that Σ∨G has an ex-treme ray of rank r.
Let R ∈ Σ∨G = S+ ∩ (ker(πG))⊥ be an extreme ray.(�) The kernel of R is a linear series on XG: ker(R) ⊂ R[x�, x�, . . . , xn]�.(�) If rk(R) > �, then ker(R) is a base-point free linear series on XG. (Blekherman)(�) �ker(R)�� is contained in R⊥.
Theorem(Blekherman, Sinn,Velasco). TheHankel indexofG isat least the (Green-Lazarsfeld index of IG) + �.
Theorem (Eisenbud, Green, Hulek, Popescu). TheGreen-Lazarsfeld index of IG isthe (smallest length of a chordless cycle inG) - �.
Positive Definite Completions
Theorem. Letm the smallest length of a chordless cycle in G. A G-partial matrixMhas a positive definite completion if and only if(�) every complete principal minor ofM is positive, and(�)M has a positive semidefinite completion of rank at least n −m + �.
MLT vs. GCR
Let G = ([n], E) be a labeled simple graph.
Definition. (�) The maximum likelihood threshold of G is the smallest wholenumber k such that there exists a positive definite matrix D with πG(M) = πG(D)for almost all positive semidefinite matricesM of rank at most k.(�) The generic completion rank of G is the smallest whole number k such thatdim(πG(Vk)) = n + �E, where Vk is the variety of matrices of rank at most k.
Proposition (Sard’sTheoremand Implicit FunctionTheorem). gcr(G) ≥mlt(G)Theorem (Blekherman, Sinn).
gcr(Km,m) = mmlt(Km,m) =min�k∶ �k + �
�� ≥ �m� ∈ O(�√m).
GramDimension (SOS length)
Definition (Laurent,Varvitsiotis). TheGramdimensionof a labeled simplegraphG = ([n], E) is the smallest whole number k such that for any positive semidef-inite matrix M ∈ S+, there exists another M′ ∈ S+ of rank at most k such thatπG(M) = πG(M′).Theorem (Laurent, Varvitsiotis). (�) k ≤ �: The Gramdimension ofG is atmost k ifand only ifG has no Kk+�minor.(�) The Gram dimension ofG is at most � if and only ifG has no K� and K�,�,�minors.
Example. The Gram dimension of anym-cylce is �.
References
Thank you
Theorem (Blekherman, Sinn).
gcr(Km,m) = mmlt(Km,m) =min�k∶ �k + �
�� ≥ �m� ∈ O(�√m).
GramDimension (SOS length)
Definition (Laurent,Varvitsiotis). TheGramdimensionof a labeled simplegraphG = ([n], E) is the smallest whole number k such that for any positive semidef-inite matrix M ∈ S+, there exists another M′ ∈ S+ of rank at most k such thatπG(M) = πG(M′).Theorem (Laurent, Varvitsiotis). (�) k ≤ �: The Gramdimension ofG is atmost k ifand only ifG has no Kk+�minor.(�) The Gram dimension ofG is at most � if and only ifG has no K� and K�,�,�minors.
Example. The Gram dimension of anym-cylce is �.
References
Thank you
Theorem (Blekherman, Sinn).
gcr(Km,m) = mmlt(Km,m) =min�k∶ �k + �
�� ≥ �m� ∈ O(�√m).
GramDimension (SOS length)
Definition (Laurent,Varvitsiotis). TheGramdimensionof a labeled simplegraphG = ([n], E) is the smallest whole number k such that for any positive semidef-inite matrix M ∈ S+, there exists another M′ ∈ S+ of rank at most k such thatπG(M) = πG(M′).Theorem (Laurent, Varvitsiotis). (�) k ≤ �: The Gramdimension ofG is atmost k ifand only ifG has no Kk+�minor.(�) The Gram dimension ofG is at most � if and only ifG has no K� and K�,�,�minors.
Example. The Gram dimension of anym-cylce is �.
References
Thank you
Theorem (Blekherman, Sinn).
gcr(Km,m) = mmlt(Km,m) =min�k∶ �k + �
�� ≥ �m� ∈ O(�√m).
GramDimension (SOS length)
Definition (Laurent,Varvitsiotis). TheGramdimensionof a labeled simplegraphG = ([n], E) is the smallest whole number k such that for any positive semidef-inite matrix M ∈ S+, there exists another M′ ∈ S+ of rank at most k such thatπG(M) = πG(M′).Theorem (Laurent, Varvitsiotis). (�) k ≤ �: The Gramdimension ofG is atmost k ifand only ifG has no Kk+�minor.(�) The Gram dimension ofG is at most � if and only ifG has no K� and K�,�,�minors.
Example. The Gram dimension of anym-cylce is �.
References
Thank you