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Williamson's theorem for positive semi-definite matrices of even size

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0

2

$begingroup$

I know the Williamson's theorem for positive definite matrices of even size. I was wondering if the theorem also holds for positive semi-definite matrices with even rank. More precisely, if $A$ is a $2n times 2n$ positive semi-definite matrix of rank $2k$ then does there exist a positive diagonal matrix $D$ of size $n$ and rank $k$ with increasing diagonal entries $0 le cdots le d_1 le cdots le d_k$ and a symplectic matrix $M$ such that $M^T A M = beginbmatrix D & 0 \ 0 & D endbmatrix$?

symmetric-matrices symplectic-linear-algebra

share|cite|improve this question

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

$endgroup$

add a comment | 

0

2

$begingroup$

I know the Williamson's theorem for positive definite matrices of even size. I was wondering if the theorem also holds for positive semi-definite matrices with even rank. More precisely, if $A$ is a $2n times 2n$ positive semi-definite matrix of rank $2k$ then does there exist a positive diagonal matrix $D$ of size $n$ and rank $k$ with increasing diagonal entries $0 le cdots le d_1 le cdots le d_k$ and a symplectic matrix $M$ such that $M^T A M = beginbmatrix D & 0 \ 0 & D endbmatrix$?

symmetric-matrices symplectic-linear-algebra

share|cite|improve this question

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

$endgroup$

add a comment | 

$begingroup$

I know the Williamson's theorem for positive definite matrices of even size. I was wondering if the theorem also holds for positive semi-definite matrices with even rank. More precisely, if $A$ is a $2n times 2n$ positive semi-definite matrix of rank $2k$ then does there exist a positive diagonal matrix $D$ of size $n$ and rank $k$ with increasing diagonal entries $0 le cdots le d_1 le cdots le d_k$ and a symplectic matrix $M$ such that $M^T A M = beginbmatrix D & 0 \ 0 & D endbmatrix$?

symmetric-matrices symplectic-linear-algebra

share|cite|improve this question

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

$endgroup$

share|cite|improve this question

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

share|cite|improve this question

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

edited Mar 14 at 4:49

J. W. Tanner

3,4601320

asked Mar 14 at 4:33

HemantHemant

1085

asked Mar 14 at 4:33

HemantHemant

1085