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Characterization of block matrices which only have positive principal minors

What are norms of sub-matrices invariant under a block diagonal similarity transformation of a block matrix?Proving Positive Definiteness of Symmetric Block MatrixIs this symmetric, block-diagonal matrix positive semi-definite?“positive matrices” in Sylvester's criterionInvertible matrices, permutations and leading principal minorsDeterminant of $3 times 3$ block matrixA Converse of Schur's product theoremIf $M$ is p.d with $M^-1=pmatrixA&B\C&D^-1=pmatrixP&Q\R&S$, then $P-A^-1$ is n.n.d.why am I getting the correct determinant?Relating the determinant of block matrix to its inverse.

0

$begingroup$

Questions

• Is there a characterization of block matrices $$ S = beginpmatrix A & B\ B^T & D endpmatrix $$ which are $P$-matrices (i.e all principal minors are strictly positive)?




• Is there a simple way to generate a (non-trivial) collection of such matrices ? The bigger / the more general the collection, the better.

N.B.: In the above display $A$ is an invertible matrix while $B$ and $D$ are rectangular matrices of appropriate shape.

Observation

By Schur's determinant formula, one has $det(S)=det(A)det(S_/A)$, where $S_/A:=D-B^TA^-1B$ is the Schur compliment of $S$.

linear-algebra matrices determinant block-matrices schur-complement

share|cite|improve this question

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

$endgroup$

add a comment | 

0

$begingroup$

Questions

• Is there a characterization of block matrices $$ S = beginpmatrix A & B\ B^T & D endpmatrix $$ which are $P$-matrices (i.e all principal minors are strictly positive)?




• Is there a simple way to generate a (non-trivial) collection of such matrices ? The bigger / the more general the collection, the better.

N.B.: In the above display $A$ is an invertible matrix while $B$ and $D$ are rectangular matrices of appropriate shape.

Observation

By Schur's determinant formula, one has $det(S)=det(A)det(S_/A)$, where $S_/A:=D-B^TA^-1B$ is the Schur compliment of $S$.

linear-algebra matrices determinant block-matrices schur-complement

share|cite|improve this question

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

$endgroup$

add a comment | 

$begingroup$

Questions

• Is there a characterization of block matrices $$ S = beginpmatrix A & B\ B^T & D endpmatrix $$ which are $P$-matrices (i.e all principal minors are strictly positive)?




• Is there a simple way to generate a (non-trivial) collection of such matrices ? The bigger / the more general the collection, the better.

N.B.: In the above display $A$ is an invertible matrix while $B$ and $D$ are rectangular matrices of appropriate shape.

Observation

By Schur's determinant formula, one has $det(S)=det(A)det(S_/A)$, where $S_/A:=D-B^TA^-1B$ is the Schur compliment of $S$.

linear-algebra matrices determinant block-matrices schur-complement

share|cite|improve this question

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

$endgroup$

share|cite|improve this question

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

share|cite|improve this question

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

edited 2 days ago

Rodrigo de Azevedo

13k41960

edited 2 days ago

Rodrigo de Azevedo

13k41960

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629

asked Feb 22 at 12:19

dohmatobdohmatob

3,682629