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Sylvester's Determinant Identity


Given a matrix A, let |A| denote its determinant. Then

 |A||A_(rs,pq)|=|A_(r,p)||A_(s,q)|-|A_(r,q)||A_(s,p)|,
(1)

where A_(u,w) is the submatrix of A formed by the intersection of the subset w of columns and u of rows. Bareiss (1968) writes the identity as

 |A|[a_(kk)^((k-1))]^(n-k-1)=|a_(k+1,k+1)^((k)) ... a_(k+1,n)^((k)); | ... |; a_(n,k+1)^((k)) ... a_(n,n)^((k))|,
(2)

where

 a_(ij)^((k))=|a_(11) a_(12) ... a_(1k) a_(1j); a_(21) a_(22) ... a_(2k) a_(2j); | | ... | |; a_(k1) a_(k2) ... a_(kk) a_(kj); a_(i1) a_(i2) ... a_(ik) a_(ij)|
(3)

for k<i,j<=n.

When k=1, this identity gives the Chió pivotal condensation method.


See also

Chió Pivotal Condensation, Determinant

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References

Bareiss, E. H. "Multistep Integer-Preserving Gaussian Elimination." Argonne National Laboratory Report ANL-7213, May 1966.Bareiss, E. H. "Sylvester's Identity and Multistep Integer-Preserving Gaussian Elimination." Math. Comput. 22, 565-578, 1968.

Referenced on Wolfram|Alpha

Sylvester's Determinant Identity

Cite this as:

Weisstein, Eric W. "Sylvester's Determinant Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SylvestersDeterminantIdentity.html

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